President Trump tweeted on Sunday that border security officials continued their "full efforts.to stop the onslaught of illegal aliens" attempting to travel to the USA southern border via a large migrant caravan - while also describing such caravans as "a disgrace to the Democrat Party".
The migrants, who said they gave up trying to enter Mexico legally because the asylum application process was too slow and most want to continue to the USA, gathered Saturday at a park in the border city of Ciudad Hidalgo.
"The caravan organizers, if they are the same as the ones who organized the caravan from the Mexican-Guatemalan border earlier this year, clearly have a political as well as a humanitarian agenda", said Theresa Brown, director of immigration and cross-border policy at the Bipartisan Policy Center and a former member of the Bush and Obama administrations.
One of the migrants marching to Ciudad Hidalgo, Luis Miguel Martinez, 30, also said his goal was to find work in the United States so he could feed his family, including two daughters he left behind in Honduras.
Some migrants used a rope to jump off the bridge and swim across the river or hitch a ride on the many rafts that cross it regularly.
The caravan of mainly Honduran migrants, whose journey has triggered escalating anti-immigrant rhetoric from US President Donald Trump, on Friday surged through a series of police lines and barricades up to the final fence on Mexico's southern border. This is the right approach: people generally don't want to leave their homes if they can live normal, safe lives there.
"UNHCR is concerned that the mobilization of such a large number of people in a single group will overwhelm the capacities that exist in the region", he told a news conference. They were not detained on reaching the Mexican bank.
Most of the migrants are citizens of Honduras, and many waved its blue-and-white national flag.
"It is a blessing that they have given us food", Martinez said.
Carrying backpacks and small children, many bedraggled migrants simply sat down on the bridge.
The soft-spoken 25-year-old said he was travelling with three friends who made a decision to remain in the caravan, but that for him, the fear of being deported once he reached Mexico was too much. On Saturday, with the punishing heat bearing down on them, some migrants took matters into their own hands, and crossed into Mexico on rafts that ferried them across the river.
While a Mexican federal police helicopter hovered overhead, hundreds of migrants remained massed on the bridge, demanding that they be allowed to cross.
Through Sunday morning, officials in Mexico had done little to stop the group from crossing into the country from Guatemala.
No one will stop us, only God.
Earlier on October 20, the presidents of Honduras and Guatemala said around 2,500 migrants were repatriated to their respective countries. Trump said on Twitter. "I hope you're not going to let these people come through your country and march a thousand miles up through your country and come through our borders, because our laws are horrendous", Mr Trump said. "They're not coming into this country".
The Mexican authorities have told migrants, who include women, children and old people, they want an orderly process and only those with valid documents will be allowed in.
The leaders said an estimated 5,400 migrants had entered Guatemala since the caravan was announced a week ago, and about 2,000 Hondurans have returned voluntarily.