On a hot and humid morning in the Mexican border city of Reynosa, less than a mile from the Rio Grande, one question seemed to linger in the minds of hundreds of people who had arrived Saturday at a shelter for migrants.

When would they be able to cross into the United States?

The answer remained elusive. At least 1,100 men, women and children, a majority of them from Central America and Venezuela, had arrived at Senda de Vida, a sprawling respite center consisting of makeshift tents and temporary wooden rooms, with hopes of reaching the United States. Instead, many felt stuck in limbo after President Biden signed an executive order that prevents migrants from seeking asylum along the 2,000-mile U.S.-Mexico border when crossings surge.

The order effectively closed the U.S. border for nearly all asylum seekers as of 12:01 a.m. on Wednesday.

The full effect of the new rule was difficult to assess three days after Mr. Biden’s announcement, but, as of Saturday, the number of migrants massing at the border showed signs of stabilizing, at least for now, compared with previous years, as many migrants appeared to be heeding the warning that they would be turned away, said Héctor Silva de Luna, a pastor who runs the shelter.

During the height of the migration crisis, he welcomed more than 7,000 people, he said. Many now appear to be waiting in the interior of Mexico, in cities like Monterrey and Mexico City, to see what happens. But the migrants at the border like the ones at Mr. de Luna’s shelter are “the ones that will pay the price,” he said, because they are being rejected.