“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.

While the migrants on the bridge waited to be processed, those who had managed to get to the Mexican side were heading to Tapachula - the next stop on a journey of at least 3,000km (1,800 miles) 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.

Mexico allows dozens of Honduran migrant women and children in, but thousands still stranded on border bridge

“We are tired, but very happy, we are united and strong,” added Edwin Geovanni Enamorado, a Honduran farmer who said he was forced to leave his country because of intimidation by racketeering gangs.

The caravan left San Pedro Sula in northern Honduras - 700km to the south - last weekend, 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 - told 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 opened the border for women and children on the overcrowded bridge, taking them to a shelter in the city of Tapachula, about 40km from Ciudad Hidalgo.

Mexican stand-off: migrant caravan stuck on bridge in Guatemala after storming border

Throughout Saturday, 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.

Morales and his Honduran counterpart Juan Orlando Hernandez said after meeting on Saturday the march was “violating the borders and the good faith of the states.”

The Honduran president admitted however 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.

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 draught, it’s for prosperity, for a better life. This is not political!”

Another migrant, who gave his name as Jaled, said they were marching “because there is no work in Honduras, no education, nothing good. The cost of life increases every weekend.”

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.