Ciudad Hidalgo, Mexico (CNN) Thousands of Central American migrants fleeing poverty and violence packed a bridge connecting Guatemala and Mexico in sweltering heat Saturday as part of a politically charged, US-bound caravan.

At least one migrant, from Honduras, was killed after falling from a vehicle, Honduran President Juan Orlando Hernandez told reporters after meeting with his Guatemalan counterpart.

"What we have seen in recent days is without precedent," he said.

At one point Saturday, several dozen migrants on the desperate trek north sang the Honduran national anthem on the edge of the muddy Suchiate River between Tecun Uman, Guatemala, and Ciudad Hidalgo, Mexico.

Central American migrants used a rope to get across part of Suchiate River.

Others, including a middle-aged woman and her three children, lowered themselves from the bridge on a ladder and climbed onto a makeshift raft that delivered them to Mexico as federal police officers watched nearby.

People from caravan wading across river now into Mexico, police don't seem to be able to stop them. pic.twitter.com/rz0c126i6W — Patrick Oppmann CNN (@CNN_Oppmann) October 20, 2018

Exhausted migrants waded across the Suchiate, holding their belongings above the chest-high waters. Rafts ferried others to Mexico.

We confirmed the Guatemala side is open if #CaravanaMigrantes wants to go back. But some are choosing to jump pic.twitter.com/DEvQSPoJoX — Bill Weir (@BillWeirCNN) October 20, 2018

A group of men helping the migrants across the river chanted "Si se puede" in Spanish, or "Yes you can." Occasionally, young men jumped from the bridge into the river as onlookers cheered.

Once in Mexico, the migrants rested at a nearby warehouse that turned into a shelter, where they received food, water and medical care.

Migrants gather at a shelter in Ciudad Hidalgo, Mexico.

Their journey continued one day after defiant caravan members rushed past border gates only to be stopped by rows of Mexican riot police who dispersed tear gas and smoke canisters into the crowds.

The slow procession north has led President Donald Trump to threaten to cut aid to Central American nations and to send troops to the US border if Mexico fails to stop the surge.

After US Secretary of State Mike Pompeo and top officials met in the Mexican capital the night before, State Department spokeswoman Heather Nauert said Saturday that Mexico was "fully engaged in finding a solution that encourages safe, secure and orderly migration."

At a rally in Elko, Nevada, Trump attacked Democrats, suggesting they want more caravans. He called the migrant situation "sad from both sides."

On the border between Mexico and Guatemala, members of the caravan seemed undeterred by rhetoric from Washington.

Honduran migrants wait for a gate on a bridge connecting Mexico and Guatemala in Ciudad Hidalgo, Mexico, to open.

"Our message is we're not criminals," Honduran migrant Orlean Herrera said. "We're coming over here because we need a better life. That's why we're here."

Another man shouted, "Donald Trump is the anti-Christ."

By Saturday morning, the bridge was teeming with men, women, teenagers and children. Some fainted from the heat.

Mexican authorities starting Friday were allowing a trickle of migrants, starting with women and children, to pass through gates near the bridge to Tapachula, where they boarded buses for refugee camps.

A Honduran migrant looks out from a gate on a bridge connecting Mexico and Guatemala in Tecun Uman, Guatemala.

Others, chanting and cheering, pushed through or climbed over a steel gate Friday until riot police repelled them with tear gas and smoke canisters. Video showed coughing and weeping migrants collapsed on the bridge. Several police and an undetermined number of migrants were injured.

It was unclear how many migrants would cross the border and how long that process would take. Mexican authorities have said they'd allow migrants to enter the country if they have valid visas or meet other requirements.

An aerial view shows the migrant caravan waiting on the Guatemala-Mexico international bridge in Ciudad Hidalgo, Chiapas state, Mexico.

The caravan formed last Saturday in San Pedro Sula, Honduras, and arrived Monday in Guatemala. Migrants told CNN en Español they were bound for the United States, seeking work and fleeing political corruption and violence.

Leaders throughout the region have publicly urged them for days to turn back.

Hernandez, the Honduran President, said more than 550 Honduran nationals were being bused back to Honduras, and would be offered jobs and other aid upon their arrival.

Pompeo: 'This is an organized effort'

Photos: The caravan's journey, in photos Migrants heading in a caravan to the United States rest on a basketball court in San Pedro Tapanatepec, Mexico. on Sunday, October 28. Hide Caption 1 of 60 Photos: The caravan's journey, in photos Migrants rest on a road between the Mexican states of Chiapas and Oaxaca after federal police briefly blocked them outside the town of Arriaga on Saturday, October 27. Hide Caption 2 of 60 Photos: The caravan's journey, in photos The caravan comes to a standstill after police block it outside Arriaga on Saturday. Hide Caption 3 of 60 Photos: The caravan's journey, in photos Police in riot gear block the highway to stop a caravan of thousands of Central American migrants from advancing through Mexico on Saturday. Hide Caption 4 of 60 Photos: The caravan's journey, in photos Migrants travel on a cattle truck, as a thousands-strong caravan of Central American migrants slowly makes its way toward the US border, between Pijijiapan and Arriaga, Mexico, on Friday, October 26. Hide Caption 5 of 60 Photos: The caravan's journey, in photos Migrants heading in caravan to the United States rest on the train tracks in Arriaga, Mexico. Hide Caption 6 of 60 Photos: The caravan's journey, in photos A truck of migrants heads in a caravan to the United States. Defense Secretary James Mattis is expected to sign deployment orders that could send 800 or more US troops to the border. Hide Caption 7 of 60 Photos: The caravan's journey, in photos Jensi, a 14 year old migrant girl from Honduras, baths in a fresh water stream as she and others, part of caravan of thousands from Central America en route to the United States, take rest in Pijijiapan, Mexico on October 25. Hide Caption 8 of 60 Photos: The caravan's journey, in photos Central American migrants rest for the night in Pijijiapan, Mexico, on Thursday, October 25. Hide Caption 9 of 60 Photos: The caravan's journey, in photos Families rest on a roadside between Mapastepec and Huixtla, Mexico, while traveling with thousands of migrants from Central America in a caravan en route to the United States, on Wednesday, October 24. Hide Caption 10 of 60 Photos: The caravan's journey, in photos Migrants ride on the back of a truck heading to the US border from the outskirts of Tapachula, Mexico, on Monday, October 22. The majority of migrants made their way on foot. But some also flagged down cars and trucks passing by and piled onto any vehicle that would take them. Hide Caption 11 of 60 Photos: The caravan's journey, in photos Migrants take shelter from the weather under a tarp as they rest in the town of Huixtla, Mexico, on October 22. Hide Caption 12 of 60 Photos: The caravan's journey, in photos Migrants select clothes that have been placed along the road by Mexican citizens. Hide Caption 13 of 60 Photos: The caravan's journey, in photos Migrants rest in the town of Huixtla, Mexico. Hide Caption 14 of 60 Photos: The caravan's journey, in photos Migrant men bathe using water from a fire hydrant at the main plaza in Tapachula, Mexico. Hide Caption 15 of 60 Photos: The caravan's journey, in photos A migrant caravan headed to the United States walks into Mexico after crossing the Guatemalan border on Sunday, October 21. Hide Caption 16 of 60 Photos: The caravan's journey, in photos Central American migrants hitchhike along the highway near the border with Guatemala, as they continue their journey trying to reach the United States. Hide Caption 17 of 60 Photos: The caravan's journey, in photos A Mexican Federal Police helicopter flies over migrants heading in a caravan to the United States, on the road linking Ciudad Hidalgo and Tapachula in Mexico. Hide Caption 18 of 60 Photos: The caravan's journey, in photos Central American migrants walk north toward Tapachula after departing Ciudad Hidalgo, Mexico. Hide Caption 19 of 60 Photos: The caravan's journey, in photos Local residents cheer as a migrant caravan walks into the interior of Mexico after crossing the Guatemalan border. Hide Caption 20 of 60 Photos: The caravan's journey, in photos Migrants still on the Guatemalan side of the border sleep at a temporary immigrant shelter on Sunday in Ciudad Tecun Uman, Guatemala. The caravan of thousands of Central Americans made its way into Mexico with some members hoping to eventually reach the United States. Hide Caption 21 of 60 Photos: The caravan's journey, in photos Honduran migrants, part of a caravan heading toward the United States, travel on the road linking Ciudad Hidalgo and Tapachula, Chiapas state, Mexico, on Sunday. Hide Caption 22 of 60 Photos: The caravan's journey, in photos Central American migrants walking to the United States continue their journey, departing Ciudad Hidalgo, Mexico, on Sunday. Hide Caption 23 of 60 Photos: The caravan's journey, in photos A Honduran migrant boy, part of a caravan heading to the United States, walks on the road linking Ciudad Hidalgo and Tapachula, Chiapas state, Mexico, on Sunday. Hide Caption 24 of 60 Photos: The caravan's journey, in photos Honduran migrants who have already reached Mexican soil cheer at the rest of the group still waiting to cross at the Guatemala-Mexico border bridge in Ciudad Hidalgo, Mexico, on Saturday, October 20. Hide Caption 25 of 60 Photos: The caravan's journey, in photos Children of migrants, part of the caravan, on Saturday wait with their parents to apply for asylum in Mexico at a checkpoint in Ciudad Hidalgo, Mexico. Hide Caption 26 of 60 Photos: The caravan's journey, in photos A Honduran migrant mother and her son walk on the bridge after crossing the border between Guatemala and Mexico on Saturday. Hide Caption 27 of 60 Photos: The caravan's journey, in photos Mexican paramedics help a Honduran woman who fainted after crossing the border between Guatemala and Mexico on Saturday. Hide Caption 28 of 60 Photos: The caravan's journey, in photos Honduran migrants help some of their members get down to the Suchiate River from the Guatemala-Mexico border bridge in Ciudad Hidalgo on Saturday. Hide Caption 29 of 60 Photos: The caravan's journey, in photos A migrant is comforted by a Mexican paramedic after her mother fainted while crossing the border between Guatemala and Mexico. Hide Caption 30 of 60 Photos: The caravan's journey, in photos Honduran migrants from the caravan on Saturday cross the Suchiate River, a natural border between Guatemala and Mexico. Hide Caption 31 of 60 Photos: The caravan's journey, in photos A Honduran migrant looks through the gate on the bridge that connects Mexico and Guatemala in Tecun Uman, Guatemala, on Saturday. Hide Caption 32 of 60 Photos: The caravan's journey, in photos Migrants in the caravan wait on Saturday for the opening of the gate on the bridge that connects Guatemala to Mexico. Hide Caption 33 of 60 Photos: The caravan's journey, in photos Thousands of Honduran migrants rush across the border toward Mexico, in Tecun Uman, Guatemala, Friday, October 19. Hide Caption 34 of 60 Photos: The caravan's journey, in photos A Honduran man protects his child after fellow migrants, part of a caravan trying to reach the United States, stormed a border checkpoint in Guatemala, in Ciudad Hidalgo, Mexico, Friday, October 19. Hide Caption 35 of 60 Photos: The caravan's journey, in photos Honduran migrants, part of a caravan trying to reach the United States, storm a border checkpoint to cross into Mexico, in Tecun Uman, Guatemala, Friday, October 19. Hide Caption 36 of 60 Photos: The caravan's journey, in photos Mexican federal police officers allow women and children taking part in a caravan of Honduran migrants heading to the United States to cross to Mexico in the border city of Tecun Uman, Guatemala, on Friday, October 19. Hide Caption 37 of 60 Photos: The caravan's journey, in photos An aerial view shows a Honduran migrant caravan heading to the United States as it is stopped at a border barrier on the Guatemala-Mexico international bridge in Ciudad Hidalgo, Mexico, on Friday, October 19. Hide Caption 38 of 60 Photos: The caravan's journey, in photos Honduran migrants heading in a caravan to the United States remove a barrier at the Guatemala-Mexico border bridge in Ciudad Hidalgo, Mexico, on Friday, October 19. Hide Caption 39 of 60 Photos: The caravan's journey, in photos A child is lifted over the border fence as thousands of Honduran migrants rush across the border toward Mexico, in Tecun Uman, Guatemala, Friday, October 19. Hide Caption 40 of 60 Photos: The caravan's journey, in photos A Honduran migrant mother and child are surrounded by Mexican Federal Police in riot gear, at the border crossing in Ciudad Hidalgo, Mexico, Friday, October 19. Hide Caption 41 of 60 Photos: The caravan's journey, in photos A Honduran migrant, part of a caravan trying to reach the United States, storms the checkpoint between Guatemala and Mexico in Tecun Uman, Guatemala, Friday, October 19. Hide Caption 42 of 60 Photos: The caravan's journey, in photos Honduran migrants heading in a caravan to the United States rush through the Guatemala-Mexico border bridge after tearing down its gate in Ciudad Hidalgo, Mexico, on Friday, October 19. Hide Caption 43 of 60 Photos: The caravan's journey, in photos Honduran migrants heading in a caravan to the United States crowd the gate of the Guatemala-Mexico border bridge in Ciudad Hidalgo, Mexico, on Friday, October 19. Hide Caption 44 of 60 Photos: The caravan's journey, in photos Honduran migrants wait at the Mexican border in Tecun Uman, Guatemala, on Friday, October 19. Thousands of migrants traveling in a caravan briefly moved toward the border crossing between the two countries before turning around. Guatemala has closed its border gate and is standing guard with dozens of troops and two armored jeeps. Hide Caption 45 of 60 Photos: The caravan's journey, in photos Honduran migrants leave Guatemala City, Guatemala, at sunrise on Thursday, October 18, 2018, as they make their way north toward the United States. Many of the more than 2,000 Hondurans in a migrant caravan trying to wend its way to the United States left spontaneously with little more than the clothes on their backs and what they could quickly throw into backpacks. Hide Caption 46 of 60 Photos: The caravan's journey, in photos Migrants display the flag of Honduras while on a caravan of migrants en route to the Mexican border on Thursday in Guatemala City. The caravan of thousands of Central Americans, most from Honduras, hopes to eventually reach the United States. President Donald Trump has threatened to cancel the recent trade deal with Mexico and withhold aid to Central American countries if the caravan isn't stopped before reaching the United States. Hide Caption 47 of 60 Photos: The caravan's journey, in photos A Honduran migrant, part of a caravan trying to reach the United States, pushes a stroller with a boy during a new leg of their travel in Guatemala City on Thursday. Hide Caption 48 of 60 Photos: The caravan's journey, in photos Honduran migrants in the caravan are pictured inside a truck on Thursday. Hide Caption 49 of 60 Photos: The caravan's journey, in photos Honduran migrants hike in the forest after crossing the Lempa River, on the border between Honduras and Guatemala, to join the caravan on Thursday. Hide Caption 50 of 60 Photos: The caravan's journey, in photos Honduran migrants climb into the bed of a truck in Zacapa, Guatemala, on Wednesday, October 17, 2018. Hide Caption 51 of 60 Photos: The caravan's journey, in photos A Honduran migrant, part of the caravan, rests on a street as he looks through his mobile phone in Guatemala City on Wednesday. Hide Caption 52 of 60 Photos: The caravan's journey, in photos A child lies on the ground to rest as Honduran migrants take part in a caravan to the United States in Teculutan, Guatemala, on Wednesday. Hide Caption 53 of 60 Photos: The caravan's journey, in photos Honduran migrants heading to the United States rest at a gymnasium of a Catholic church in Chiquimula, Guatemala, on Tuesday, October 16, 2018. The caravan set out October 13 from the impoverished, violence-plagued country. Hide Caption 54 of 60 Photos: The caravan's journey, in photos Honduran migrants pray at an improvised shelter in Chiquimula, Guatemala, on Tuesday. Hide Caption 55 of 60 Photos: The caravan's journey, in photos Families in the caravan rest for the night in a community gym on Tuesday in Chiquimula, Guatemala. The caravan is the second of its size in 2018. Hide Caption 56 of 60 Photos: The caravan's journey, in photos Honduran migrants move north on Tuesday near Quezaltepeque, Guatemala. Hide Caption 57 of 60 Photos: The caravan's journey, in photos Honduran migrants holds up their national ID cards as Guatemalan police block them and their caravan after the group crossed the Honduras-Guatemala border in Esquipulas, Guatemala, on Monday, October 15, 2018. Police stopped the migrants for several hours but the travelers refused to return to the border and were eventually allowed to pass. Hide Caption 58 of 60 Photos: The caravan's journey, in photos The caravan moves north after crossing the border from Honduras into Guatemala on Monday. Hide Caption 59 of 60 Photos: The caravan's journey, in photos The caravan pauses at a Guatemalan police checkpoint after crossing the border from Honduras on Monday. Hide Caption 60 of 60

Four Mexican police officers were injured in the standoff at the Mexican border, Pompeo said Friday.

Speaking to reporters on a tarmac in Mexico City, where he'd been meeting with leaders about the caravan and other matters, Pompeo accused the caravan of using women and children "as shields as they make their way through."

"This is an organized effort to come through and violate the sovereignty of Mexico," Pompeo said.

Pompeo said he had a "good conversation" with Mexican President Enrique Peña Nieto and that the two countries are jointly focused on the caravan issue, but "the Mexican government is making all the decisions on how to address this."

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Peña Nieto called the confrontation a "unique incident," saying the migrants did not follow the protocols that Mexican immigration officials put in place. He added that some migrants "assaulted and even injured" a number of unarmed members of the Mexican police before they could contain the violence.

"Mexico won't allow anyone entering its territory in an erratic way and much less a violent way," he said Friday night in a video statement released on Twitter.

Peña Nieto said Mexican officials would continue welcoming those traveling with the migrant caravan despite Friday's incident.

Trump, who's been criticizing the caravan with fiery tweets all week, reiterated his threat Friday that if Mexico doesn't stop the migrants, he'll take more drastic action.

"If that doesn't work out, we're calling up the military, not the Guard, we're calling up the military, and we're going to have the military stationed," Trump said. "They're not coming into this country. They might as well turn back."

Trump has also threatened to cut foreign aid to Honduras, Guatemala and El Salvador.

Clash on the bridge

Members of the caravan gathered Friday morning in the border city of Tecun Uman. They waited in sweltering heat -- with temperatures in the mid-80s and 70% humidity -- for hours for others to arrive, packing streets for blocks on end.

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They streamed through gates on the Guatemalan side of the border around noon Friday, cheering and chanting as they marched. Celebratory air horns blared as they headed toward Mexico's port of entry.

When they reached a padlocked steel fence near the Mexican side of the bridge, the sheer number of migrants allowed them to break through.

Mexican police pushed them back. Hundreds of Mexican Federal Police sealed the border as drones and helicopters hovered above the crowd.

At least a dozen migrants jumped into the river.

Integrantes de la Caravana de hondureños se arroja al río suchiate para intentar seguir su camino. Al fondo Guatemala. pic.twitter.com/SfuZ0yxZi2 — Krupskaia Alís (@AlisCNN) October 19, 2018

Some of the migrants on the bridge said they joined the caravan because they were desperate for work and had no other choice.

Mexican officials have said people seeking asylum will be processed at Mexico's southern border. But it's unclear what will happen next

Will they reach the US border?

Mexican authorities haven't said whether anyone from the caravan will be permitted to travel to the US border.

They've previously outlined how they planned to respond to the group, stating:

• Anyone with a valid visa will be able to enter and move freely.

• Anyone who wants to be recognized as a refugee or as a beneficiary of "complementary protection measures" must do so individually. Those who do so will be held "at a migratory station" for as many as 45 business days.

• Anyone who enters "in an irregular manner" will be "rescued and subject to an administrative procedure and, where appropriate, will be returned to their country of origin in a safe and orderly manner."

This spring, when another caravan of Central American migrants crossed into Mexico, such policies allowed that group to make it to the US border.

The latest caravan comes weeks before high-stakes midterm elections in the United States, in which many Republican candidates have been echoing the President's messaging about boosting border security and cracking down on illegal immigration.

Trump has pointed to the caravan as a key issue in the upcoming elections, describing the group's approach as an assault on the US border. Immigrant rights advocates have accused his administration of trying to create a crisis to motivate his voter base.

Honduran and Guatemalan leaders talk

Hernandez met Saturday in Guatemala with his counterpart, Jimmy Morales, to work on "implementing a strategy and logistics for the return of Honduran migrants," Hernandez said on Twitter.

Hernandez also sought permission from Guatemala to send Civil Protection personnel to assist Honduran migrants inside Guatemala on Friday, according to several posts on his official Twitter account.

In addition, Hernandez requested authorization to "hire ground transportation for those wishing to return, even airlifting special cases of women, children, the elderly and sick," he said, thanking the Guatemalan government for "all the attention given to our Honduran brothers."