TIJUANA, Mexico—When a group of nearly 200 migrants from Central America reached the U.S. border crossing here Sunday, U.S. Customs and Border Protection said the crossing was full and couldn’t process asylum seekers.

The migrants are part of a caravan that has drawn the attention, and ire, of President Donald Trump. The president has said the group’s plan to ask for asylum is evidence that there is a crisis at the U.S. border with Mexico.

The...