TECUN UMAN, Guatemala—A caravan of several thousand Honduran migrants on their way to the U.S. breached Mexico’s southern border with Guatemala on Friday, but were held at bay by hundreds of Mexican police in a tense standoff, deepening a regional migration crisis.

The rush at the border came after Mexican officials said they would grant asylum to the caravan of about 3,000 migrants fleeing violence-prone Central America. But the Mexicans said they would let in only 100 to 200 migrants a day, because they couldn’t process...