Mexican authorities had managed to block the "caravan" of migrants on a border bridge between Mexico and Guatemala, but many later crossed the river below in makeshift rafts before marching north.
An irate Trump insisted that "full efforts" were under way to halt the caravan’s progress toward the United States.
"Full efforts are being made to stop the onslaught of illegal aliens from crossing our Souther(n) Border," Trump tweeted.
"People have to apply for asylum in Mexico first, and if they fail to do that, the US will turn them away."
Nevertheless, around 3,000 people were marching in the caravan on the Mexican side, according to an estimate from a federal police commander whose forces were closely monitoring the migrants’ progress.
About a thousand migrants, including women and children, were still stranded on a border bridge hoping to enter Mexico legally via Guatemala.
Mexican authorities insisted those on the bridge would have to file asylum claims one at a time in order to enter the country.
And another separate group of about 1,000 Hondurans started their own march across Guatemala, headed for Mexico and then the United States. The group of men, women and children gathered in Esquipulas before setting out on foot.
’No stopping us’
After seven long hours walking in heat and humidity, the larger group on the Mexican side made it to Tapachula as part of their journey of at least 3,000 kilometres to the border between Mexico and the United States.
"No one is going to stop us, after all we’ve gone through," said 21-year-old Aaron Juarez, who was accompanied by his wife and baby and was walking with difficulty because of an injury.
Honduran farmer Edwin Geovanni Enamorado said he was forced to leave his country because of intimidation by racketeering gangs.
"We are tired, but very happy, we are united and strong," he said.
Britany Hernandez added: "We have sunburn. We have blisters. But we got here. Our strength is greater than Trump’s threats."
Mexico’s President-elect Andres Manuel Lopez Obrador called for fair treatment of the migrants.
"We don’t want them to face what (Mexicans) face when they need to look for work in the United States," he said on Twitter.
The caravan left San Pedro Sula in northern Honduras more than a week ago, following a call on social networks relayed by a former Honduran deputy.
The politician, Bartolo Fuentes -- a member of leftist former president Manuel Zelaya’s Freedom and Refoundation Party – said he only reproduced a poster on his Facebook page.
The poster invited people on a "Migrant march" with a slogan: "We’re not leaving because we want to, but because we are being expelled by violence and poverty."
The caravan has comprised between 3,000 - 5,000 people at various times as it moved through Guatemala, according to various sources.
Guatemalan President Jimmy Morales said more than 5,000 migrants had entered Guatemala from Honduras, but that some 2,000 had since returned home.
Officials of the national disaster management agency, CONRED, said that more than 1,000 Hondurans had left the caravan between Friday and Sunday, taken home on a fleet of buses laid on by the Guatemalan government.
On Saturday, Mexican authorities had opened the border for women and children on the overcrowded bridge, taking them to a shelter in the city of Tapachula, about 40 kilometers from Ciudad Hidalgo.
Plenty of migrants, however, are sleeping in the streets for fear that immigration officials could arrest them if they are in a shelter.
Rafting the River
Around 900 migrants -- tired of waiting on the bridge -- resorted to crossing the Suchiate River below on makeshift rafts and police did not intervene as they clambered up the muddy riverbank on the Mexican side on Saturday.
Morales and his Honduran counterpart Juan Orlando Hernandez said after meeting that the march was "violating the borders and the good faith of the states."
The Honduran president acknowledged that social problems were a contributory factor.
"Without a doubt, we have a lot to do so that our people can have opportunities in their communities," he said.
Migrants denied their motives were political.
"We decided to join those who were going," said Edgar Aguilar.
"This is not political. This comes from hunger, from the drought, it’s for prosperity, for a better life. This is not political!"
The migrants are generally fleeing poverty and insecurity in Honduras, where powerful street gangs rule their turf with brutal violence.
With a homicide rate of 43 per 100,000 citizens, Honduras is one of the most violent countries in the world, according to a Honduran university study. — AFP/VNS