Secretary of State Mike Pompeo raised the stakes Sunday night as thousands of Central American migrants headed for the United States marched into Mexico, claiming they were inciting violence under the influence of politically motivated leaders.

There's no precise calculation of how many migrants make up the caravan, but Mexican civil protection authorities said Sunday that about 7,000 people, most of them from Honduras, had set out from an area near the Mexican-Guatemalan border for Tapachula in the Mexican state of Chiapas.

About 2,000 others have been intercepted and returned to Honduras, Guatemalan President Jimmy Morales said.

The United States has put pressure on Mexico and Honduras to stop the procession, warning over the last week that the travelers were vulnerable to human smuggling rings and other criminal organizations.

Honduran migrants taking part in a caravan to the United States wait to cross to Mexico in Ciudad Tecun Uman, Guatemala, on Saturday. Orlnado Sierra / AFP - Getty Images

On Sunday night, Pompeo — who met with Mexican President Enrique Peña Nieto and Mexican Foreign Secretary Luis Videgaray Caso — went on the offensive, alleging in a statement that many of the migrants were criminals themselves.

"We also are deeply concerned by the violence provoked by some members of the group, as well as the apparent political motivation of some organizers of the caravan," Pompeo said.

"As President Trump has stated, consistent with U.S. law, the United States will not allow illegal immigrants to enter or remain in the United States," said Pompeo, who offered no examples of violence instigated by the migrants.

The travelers have said they are fleeing unemployment and violence in Honduras.

Pueblo Sin Fronteras, which has led caravans of migrants to the United States for more than 15 years, accused the Mexican and Guatemalan governments of using fear and racism to do Washington's "dirty work," saying in a statement on Sunday that they were using "excessive violence" to stop the group.

"We've already seen that the Mexican government has bent under the pressure of the U.S. government and that it has become an accomplice to the illegitimate, repressive government of Juan Orlando Hernández," the president of Honduras, Pueblo Sin Fronteras said.

On Friday, six Mexican federal police officers were injured in an altercation with migrants along the border, Mexican authorities said. The country's National Migration Institute on Sunday urged the travelers to head to shelters and to apply to stay in Mexico legally, but many have said they plan to continue marching north toward the United States on Monday.

Trump, who has previously threatened to cut off all aid to Honduras if the caravan isn't stopped, said Sunday on Twitter that the procession was "a disgrace to the Democrat Party," without offering evidence of Democratic involvement.

"People have to apply for asylum in Mexico first, and if they fail to do that, the U.S. will turn them away," the president tweeted.