"It was a great sacrifice, but it's all for a better life", he said.
In part, the response is meant to both dissuade the migrants from even attempting to enter Mexico and potentially to show the Trump administration that Mexico is willing to, as the government put it, "maintain order".
He called on Mexico to ensure the migrants do not reach the United States border while threatening Guatemala, Honduras and El Salvador with financial penalties.
Authorities in Ciudad Hidalgo, on the Mexican side, were expecting around 3,000 migrants to arrive in total.
A federal police helicopter circled overhead while observers from the national Human Rights Commission were sent to various points along the 900-kilometer (550 miles) border.
The group of Central American migrants has made its intentions clear: they are heading to the US. He said they were "worried that things could escape rational margins". Others could opt for the rafts that ferry people - and historically migrants - daily across the river. "Now we're being pushed back", Weir said.
Dozens of Mexican federal police officers were on the bridge, backed up by hundreds more officers deployed behind it, and Mexico's ambassador to Guatemala said his country was enforcing what he called a policy of orderly entry in the face of the thousands trying to cross. Mujica appeared to be resisting.
Trump and his administration are working toward the reimplementation of a version of the family separation policy enacted last spring, when thousands of immigrant children were taken from their parents and loved ones. It said Mujica attacked immigration agents, as well as local and federal police, after he was asked for his identification. They also said migrants caught without papers would be deported.
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There - at the far end of the bridge over the Suchiate River, which forms the western part of the Mexico-Guatemala border - they hurled rocks and other objects at hundreds of riot police, who responded with rubber bullets and tear gas.
The president's threats also come just weeks before the mid-term elections on 6 November, which could see Democrats unseating Republicans on Capitol Hill.
He blamed Democrats for obstructing his attempts at reforming the country's immigration laws. Trump said in one tweet.
Pompeo met his Mexican counterpart, Luis Videgaray, and the caravan was the issue that preoccupied them.
"Nobody likes them (Trump's comments)".
Mexico has requested the assistance of the United Nations in the process of issuing asylum and deportation. "And the step we took today, it's extremely important".
Mexico says the migrants will be processed and that those without a legitimate case to travel onward or stay in Mexico will be returned to their countries of origin.