Honduran migrants, part of a caravan trying to reach the U.S., gesture while arriving to the border between Honduras and Guatemala, in Agua Caliente, Guatemala October 15, 2018.

U.S. President Donald Trump on Tuesday said he has warned the president of Honduras that if the caravan of migrants heading to the U.S. is not brought back to Honduras, no more aid will be given to Honduras, "effective immediately."

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Hundreds of Hondurans hoping to reach the U.S. bedded down for the night in this Guatemalan town after that country's authorities blinked first in attempts to halt their advance.

The group estimated at 1,600 to 2,000 people fleeing poverty and violence in Honduras marched into Guatemala in sweltering heat Monday, twice pushing past outnumbered police sent to stop them — first at the border and then at a roadblock just outside Esquipulas.

After those encounters, Mexico's immigration authority sent out a fresh warning late Monday that the migrants would have to satisfy Mexican officials individually and that only those meeting requirements would be allowed to enter.

U.S. authorities were watching as well. Katie Waldman, a U.S. Department of Homeland Security spokeswoman, said in a statement that the caravan was "what we see day-in and day-out at the border as a result of well-advertised and well-known catch-and-release loopholes."

"Until Congress acts, we will continue to have de-facto open borders that guarantees future 'caravans' and record numbers of family units entering the country illegally," she said.

The exhausted migrants entered Esquipulas during the evening and sought out food and places to sleep, hobbling on blistered feet. Few carried food and some residents began to organize to help feed them. Some migrants asked for money, others passing a bakery were handed bread.

The migrants arrived at the Guatemalan border singing the Honduran national anthem, praying and chanting, "Yes, we can." The group defied an order by the Guatemalan government that they not enter.

"We have rights," the migrants shouted.

Keilin Umana, a 21-year-old who is two months pregnant, said she was moved to migrate to save herself and her unborn child after she was threatened with death.

Umana, a nurse, said she had been walking for four days. "We are not criminals — we are migrants," she said.

Many in the caravan traveled light, with just backpacks and bottles of water. Some pushed toddlers in strollers or carried them on their shoulders.

Carlos Cortez, a 32-year-old farmer traveling on foot with his 7-year-old son, said poverty back home made it impossible to support a family.

"Every day I earn about $5," Cortez said. "That isn't enough to feed my family."

The caravan was met at the border by about 100 Guatemalan police officers. After a standoff of about two hours, the migrants began walking again. Officers did nothing to stop them, but accompanied them several miles (kilometers) into Guatemalan territory.

Officers then set up the roadblock about a mile (2 kilometers) outside Esquipulas. About 250 police kept them from advancing for three hours, telling them they had to return to the border to go through immigration. The migrants refused to budge and eventually officers again let them pass.

The caravan began as about 160 people who first gathered early Friday to depart from San Pedro Sula, one of Honduras' most dangerous places, figuring that traveling as a group would make them less vulnerable to robbery, assault and other dangers common on the migratory path through Central America and Mexico.

Local media coverage prompted hundreds more to join during the weekend as the group moved toward Guatemala.

A day before the caravan formed, U.S. Vice President Mike Pence had urged leaders in Honduras, El Salvador, and Guatemala to persuade their citizens to stay home and avoid the long, risky journey to the United States.

In April, Trump threatened to withdraw foreign aid from Honduras and countries that allowed transit for a similar caravan that set out from the Central American country. That caravan dwindled as the group approached the U.S. border, with some giving up along the way and others splitting off to try to cross on their own.

Historian Dana Frank, an expert on human rights and U.S. policy in Honduras, said Monday that the latest group could have political implications in the United States with the midterm elections coming up.

Frank said that "some in the United States will be quick to raise alarms about a supposed dangerous immigrant invasion" and that "others will view these migrants with compassion and as further evidence of the need for comprehensive immigration reform."

Frank said the caravan's rapid growth underscores "how desperate the Honduran people are — that they'd begin walking toward refuge in the United States with only a day pack full of belongings."

Honduras is largely dominated by murderous gangs that prey on families and businesses, and routinely sees homicide rates that are among the highest in the world.