The caravan of mainly Honduran migrants had surged through a series of police lines and barricades up to the final fence on Mexico´s southern border on Friday.
US Department of Homeland Security Secretary Kirstjen Nielsen noted that her agency is monitoring the caravan while going so far as to suggest that the migrant movement can serve as a cover for transnational criminal organizations.
"I'm looking for a better future for my children".
She said she had waited at the border until mid-Saturday but became desperate at how long it was taking to be processed.
A growing group of Central American migrants resumed their advance toward the USA border in southern Mexico on Sunday, overwhelming Mexican government attempts to stop them at the border.
Those restrictions were too onerous for many migrants, for whom Mexico is a hurdle to be surmounted as they reach for their real goal of taking jobs in the USA or reuniting with family already in the country - usually illegally.
"The United States has strongly informed the President of Honduras that if the large Caravan of people heading to the U.S.is not stopped and brought back to Honduras, no more money or aid will be given to Honduras, effective immediately!"
Hundreds slept in the open on the bridge over the Suchiate River and in the main square of the Guatemalan border town of Tecun Uman.
On Saturday the Honduran Office of Foreign Ministry released a statement noting that it had closed the border with Guatemala at Aguas Calientes in order to "save the lives and physical integrity of citizens and foreigners" that try to pass through customs in this city.
Civil defense officials for Mexico's southern state of Chiapas said they had offered to take the migrants by bus to a shelter set up by immigration officials about 5 miles (7 kilometers) outside Tapachula, but the migrants refused, fearing that once they boarded the buses they would be deported.
"We are one group", he said.
Mr Trump has threatened to halt aid to Honduras and Guatemala, and potentially close the U.S. border with Mexico with the help of the military if the migrants' march is not stopped.
A group of Central American migrants cross the Suchiate River aboard a raft, on the the border between Guatemala and Mexico, in Ciudad Hidalgo, Mexico on Saturday. 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. Mexican authorities previously outlined their plan to respond to the caravan once it arrived at the Mexican border.
"They're not coming into this country", said Trump, who has sought to make the caravan and border security in general a campaign issue for Republicans ahead of the USA midterm elections.
Numerous migrants hail from Honduras, fleeing violence, poverty and a lack of jobs.
Anthropologist Frank-Vitale adds that a caravan out of El Salvador is already in the works: "Inspired by Honduras there are rumored caravans to begin out of El Salvador on Oct 31".
The caravan formed just weeks before the USA midterm elections. "I hope they continue". Trump said on Twitter. But as of this moment, I thank Mexico.
'We are not criminals, we are workers!'
"They might as well turn back, they're not coming into this country", he told reporters on Friday.
"We fully support the efforts of Guatemala, Honduras, and Mexico, as they seek to address this critical situation and ensure a safer and more secure region", she added.
After an emergency meeting in Guatemala, presidents Hernandez of Honduras and Jimmy Morales of Guatemala 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.
The migrant rights group Pueblos Sin Fronteras also counted thousands of mostly Honduran migrants nearby Cuidad Hidalgo, although the figures did not exactly match.