"In addition to stopping all payments to these countries, which seem to have nearly no control over their population, I must, in the strongest of terms, ask Mexico to stop this onslaught - and if unable to do so I will call up the US Military and CLOSE OUR SOUTHERN BORDER!" Trump wrote on Twitter.
Earlier Friday, a New York Times reporter challenged Trump's assertion that migrants in the caravan were criminals.
Mexico has now requested that the United Nations High Commissioner for Refugees establish shelters along the border with Central America, a move both confirmed and welcomed by U.S. Secretary of State Mike Pompeo.
Mexican president-elect Andrés Manuel López Obrador plans to offer work visas to Central Americans when he takes office in December.
As Trump said during his recent "60 Minutes" interview, "When you allow the parents to stay together, OK, when you allow that, then what happens is people are gonna pour into our country".
As the sun rose, a military helicopter could be seen overhead, foreshadowing the difficulties that migrants will have as they try to reach the U.S.
Cristian, who is among the informal leaders of the caravan, estimated about 30 percent of the migrants want to apply for refugee status in Mexico.
The caravan of almost 3,000 people left San Pedro Sula in Honduras late last week, making its way through Guatemala's muddy jungle and residential streets to reach Mexico and, eventually, the United States.
But foreign minister Videgaray said that was "false" and that no such deal had been made. During a presidential debate, Lopez Obrador indicated his administration would stop doing "dirty work" for the United States by detaining Central American migrants.
"The journey has been very tough, very hard but there's no work in Honduras", 20-year-old Glenda Salvador, a mother of two toddlers, told AFP earlier in the day as the migrants massed in the Guatemalan town of Tecun Uman, close to the bridge. The nearest border is about 110 miles (177 km) away.
"Well, we're looking at a lot of things", he said.
Arriving on the Mexican side of a border bridge, they were met by a phalanx of police with riot shields.
Vaughan stressed that it is a good sign that Mexico is realizing that it has more to gain than lose by cooperating with the US on border enforcement. They also said migrants caught without papers would be deported.
"Mexico does not and will not allow (people) to enter its territory illegally, let alone violently", he said. More than 150 members of that group crossed the border and sought asylum.
Mexico's southern border is notoriously porous and it was unclear how numerous migrants would attempt to cross legally at the bridge.
He was referring to the United States-Mexico-Canada Agreement, which is awaiting ratification. "Great Midterm issue for Republicans!"
"We have to secure that border once and for all", McCaul told Fox News. There are no official estimates of the size of the group.
On Friday, Scalise, who survived a politically-motivated shooting in 2017, said "it's obvious" Trump was not "encouraging his supporters to engage in attacks" during his Thursday night rally. Even if it's within his authority as president, he would nearly certainly face stiff resistance, obstruction, and nigh unending court battles over it - much the same response he might expect, say, for deploying the military to enforce a closed border.