SAN LUIS RÍO COLORADO, Mexico—The U.S. has slashed the number of migrants being allowed to cross the U.S.-Mexican border to legally apply for asylum, as caravans totaling some 10,000 migrants trudge north through Mexico.

Here at the border station across from Yuma, Ariz., 30 families or more normally cross each day, say Mexican immigration officials. But in the past two weeks the U.S. Customs and Border Protection has let in one family a day at most, migrants and their advocates say.

“It really seems like they are trying to discourage people from crossing to seek asylum legally, or trying to get them to go to other border crossings,” said Iveth López, an immigration counselor with Chicanos Por La Causa in Somerton, Ariz. “It’s been bad for a couple of weeks.”

Reports of slowdowns at legal border crossings like this one are becoming commonplace across the 2,000-mile frontier from Texas to California. Immigration lawyers have complained of asylum seekers being blocked in recent days from entering the U.S. from the Mexican border cities of Matamoros, Reynosa and Ciudad Juárez as well.

Last week, U.S. authorities announced they would “harden” the border at Tijuana, closing four lanes of traffic at the San Ysidro and Otay Mesa crossings. As some 1,000 migrants arrived in Tijuana one recent day, border-patrol contractors unwound hundreds of yards of razor wire at the beachside border fence, as armed guards looked on.