At the shelter in San Diego, only nationals from countries like India, China and Russia, whose citizens are not subjected to the Remain in Mexico policy, continue to trickle in, Ms. Clark said. The only exceptions to the policy among Latin Americans are those who have a child or another family member with a serious medical issue. A Honduran man who identified himself only as Jonnie arrived with his 4-year-old girl, who had advanced scoliosis. He said she was in critical need of surgery.

Elsewhere along the Mexican side of the border, there were also signs that fewer migrants were arriving from the south.

In Nuevo Laredo, on the Mexican side of the border opposite Laredo, Tex., the number of residents in the half-dozen migrant shelters in the city has dropped sharply in recent weeks, after months of operating well above capacity. At the Casa del Migrante Nazareth, the number of migrants, nearly all of them intending to seek asylum in the United States, has hovered around 100 in recent days, well below the peak of 290 at the beginning of May.

The Casa del Migrante Amar, another shelter in the city, was housing about 100 migrants on Wednesday, about a fifth of the number that crammed the center for weeks earlier this year.

Shelter operators said there could be several reasons for the drop, including stricter occupancy limits imposed by the shelters, and the city’s reputation for being a dangerous place, which has perhaps compelled migrants to head to other border cities. But some advocates here also said it was possible that the joint Mexico-United States crackdown on undocumented migration could be slowing the flow.

“It could be dissuading migrants,” Father Julio López, who runs the Casa del Migrante Nazareth, said of the new bilateral strategies. They might be biding their time in their home countries or elsewhere on the migrant trail, he said, “waiting to see what will happen.”