He even blamed the Democrats for the caravan - while offering no evidence.
The group's decision 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 of them. "We're wet and we still don't have a place to sleep".
Meanwhile, a Honduran caravan of migrants headed for the U.S. border has grown to 4,000.
The Mexican government said in a statement that it had received "requests for refuge" from 640 migrants, including 104 minors.
The U.S. president has made it clear to Mexico that he is monitoring its response.
At a news conference with Videgaray, Pompeo called illegal migration a "crisis" and emphasized "the importance of stopping this flow before it reaches the United States border", while also acknowledging Mexico's right to handle the crisis in a sovereign fashion. The majority seek to use the country's northern border to infiltrate the USA illegally, according to the Associated Press.
A 12-year-old boy named Mario David Castellanos Murillo was arrested by Mexican police on the global bridge yesterday after making the journey from Honduras by himself. Metal barricades were stored to one side, but not yet deployed.
This family spent the night of 19 October at a temporary shelter in Ciudad Hidalgo, Mexico.
"We hope that the immigration officials and federal police have a humanitarian understanding", Corzo said. Efforts to retrieve them were ongoing, authorities said.
Police officers patrolling the area told AFP that for now there are no plans to move against the migrants crossing by river. Others could opt for the rafts that ferry people - and historically migrants - daily across the river.
Some migrants violently shook fences at the border and police said a few officers were injured in clashes.
"This migration has political motivations", said Morales, "which is violating the borders and the good faith of the states and of course putting at risk the most important thing, people".
Corzo said police accused Mujica of slashing the tires of an immigration vehicle.
Pompeo said he and Videgaray spoke of the importance of stopping the caravan before it reaches the U.S. border. -Mexican citizenship, is accused of property damage. They also said migrants caught without papers would be deported.
Honduran President Juan Orlando Hernandez said he would meet Saturday with his Guatemalan counterpart Jimmy Morales to discuss the next steps in the crisis, which has triggered escalating anti-immigrant rhetoric from US President Donald Trump.
Mr Trump's threats come weeks before the mid-term elections on 6 November, which could see Democrats unseating Republicans in the House of Representatives.
President Trump has warned them that he will shut down the southern USA border so they can not enter.
Trump's stance, he said, was "what he has always presented", adding he saw "nothing surprising in it".
"If that's the case, I don't think Mexico should be too anxious because in a sense ... it's the same kind of thing USA administrations have been doing for a long time", Leutert said.
A vigorous immigration push will likely be well-received in numerous deep-red areas where Trump is campaigning, like his stop in Montana Thursday night. The migrants travel in groups for safety.
On Friday, they broke down Guatemalan gates and streamed toward a bridge to Mexico.
Caravan participants are escaping brutal conditions of poverty and state violence, enforced by American imperialism.
Elizabeth Oglesby, a professor at the University of Arizona's Center for Latin American Studies, said people join caravans like this because it's a way to make the journey in a relatively safe manner and avoid having to pay thousands of dollars to smugglers. They extort them for money", Oglesby said, "They sometimes force them to work as couriers for drug trafficking.
They left Honduras just days after Central American leaders met with US Vice President Mike Pence. "We are not going to attend to the issue only with deportations or means of force".