Mexico responded. This week, when Mexican and American officials met in Washington to discuss progress on the issue, the Mexican delegation took great pains to show how its crackdown along its border with Guatemala and throughout the country has reduced migration flows to the United States by more than 50 percent in the last three months.

Mexico’s actions, though applauded by Trump administration officials this week, have overwhelmed its troubled migration system. The number of individuals applying for asylum in Mexico has already skyrocketed in the last few years, as the United States has tightened its borders.

This rule could add to that burden, with many more applying for asylum in Mexico, despite the danger of remaining in Mexico. Violence there has soared to the highest levels in more than two decades. Stories of migrants kidnapped along the border abound, as criminal organizations await their return from the United States or pick them off as they attempt to cross the border.

Several migrants who are making their way north said in interviews on Thursday that the new rule would not deter them. For most, the hope of a new life in the United States outweighed whatever legal worries might lie ahead.

“I know things are getting more and more complicated in the U.S.,” said Noel Hernández, 21, who was staying at a migrant shelter in Guatemala after leaving his home in Tegucigalpa, in Honduras, a few days ago.

“It’s like flipping a coin,” he said. “I either win or I lose.”

Others said they would try to make it in Mexico, despite the violence, or in Guatemala, a nation with a barely functional asylum system.