A growing number of refugees from Central America have resumed their advance towards the US border in southern Mexico, despite Mexican efforts to stop them at the border with Guatemala.

The “caravan” swelled to about 5,000 overnight and they set out walking at first light on Sunday toward the Mexican town of Tapachula, 10 abreast in a line stretching back approximately a mile.

About 2,000 people had gathered on the Mexican side on Saturday night. The newcomers appear to be drawn from people who who had been waiting on the bridge over the Suchiate River or in the Guatemalan town of Tecun Uman and who decided to cross during the night. At dawn, there were still an estimated 1,500 people on the Guatemalan side hoping to enter legally.

On Saturday, many of those waiting to cross the border officially became impatient and circumvented the border gate by crossing the river on rafts, by swimming or by wading in full view of the hundreds of Mexican police at the blockade on the bridge. Some paid locals the equivalent of $1.25 to ferry them across the muddy waters. They were not detained on reaching the Mexican bank.

Play Video 0:35 Thousands of Hondurans continue marching towards US – video

As the caravan passed through Mexican villages on the outskirts of the border city of Ciudad Hidalgo on Sunday, they drew applause, cheers and donations of food and clothing from Mexicans.

Olivin Castellanos, 58, a truck driver and mason from Villanueva, Honduras, said he took a raft across the river after Mexico blocked the bridge. “No one will stop us, only God,” he said. “We knocked down the door and we continue walking.” He wants to reach the US to work. “I can do this,” he said, pointing to the asphalt under his feet. “I’ve made highways.”

The refugees, who said they gave up trying to enter Mexico legally because the asylum application process was too slow, gathered at a park in Ciudad Hidalgo. They voted by a show of hands to continue north en masse, then marched to the bridge crossing the Suchiate and urged those still on it to join them.

The decision to reassemble the refugee caravan capped a day in which Mexican authorities again refused mass entry to migrants on the bridge, instead accepting small groups for asylum processing and giving out 45-day visitor permits to some.

Sairy Bueso, a 24-year old Honduran mother of two, abandoned the bridge and crossed into Mexico via the river. She clutched her two-year-old daughter Dayani, who had recently had a heart operation, as she got off a raft.

“The girl suffered greatly because of all the people crowded on the bridge,” Bueso said. “There are risks that we must take for the good of our children.”

In addition to those who crossed the river, immigration agents processed migrants in small groups and then took them by bus to an open-air, metal-roof fairground in Tapachula, where the Red Cross set up small blue tents on the concrete floor.

Facebook Twitter Pinterest Honduran migrants wait to cross to Mexico. Photograph: Orlando Sierra/AFP/Getty Images

Mexico’s interior department said it had received 640 refugee requests by Hondurans at the border crossing. It released photos of migrants getting off buses at a shelter and receiving food and medical attention. At least half a dozen migrants fainted in the crush.

People in the caravan cited widespread poverty and gang violence in Honduras, one of the world’s deadliest nations by homicide rate, as their reasons for fleeing.

Juan Carlos Mercado, 20, from Santa Barbara, said corruption and a lack of jobs in Honduras had stymied him. “We just want to move ahead with our lives,” he said, adding that he would do any kind of work.

The caravan elicited a series of angry tweets and warnings from Donald Trump early in the week, but Mexico’s initial handling of the migrants at its southern border seemed to have satisfied him.

“So as of this moment, I thank Mexico,” Trump said Friday at an event in Scottsdale, Arizona. “I hope they continue. If that doesn’t work out, we’re calling up the military not the [National] Guard. They’re not coming into this country.”