TIJUANA, Mexico — More buses of exhausted people in a caravan of Central American asylum seekers reached the U.S. border Thursday.

The city of Tijuana converted a municipal gymnasium into a temporary shelter, as the migrants came to grips with the reality that they will be on the Mexican side for an extended stay.

With U.S. border inspectors at the main crossing into San Diego processing only about 100 asylum claims a day, it could take weeks if not months to process the thousands in the caravan that departed from San Pedro Sula, Honduras, more than a month ago.

Tijuana’s robust network of shelters was already stretched to the limit, having squeezed in double their capacity or more as families slept on the floor on mats, forcing the city to open the gymnasium for up to 360 people Wednesday. A gated outdoor courtyard can accommodate hundreds more.

The city’s thriving factories are always looking for workers, and several thousand Haitian migrants who were turned away at the U.S. border have found jobs and settled here in the last two years, but the prospect of thousands more destitute Central Americans has posed new challenges.

Delia Avila, director of Tijuana’s family services department, who is helping spearhead the city’s response, said migrants who can arrange legal status in Mexico are welcome to stay.

“Tijuana is a land of migrants. Tijuana is a land that has known what it is to embrace thousands of co-nationals and also people from other countries,” Avila said.

To claim asylum in San Diego, migrants enter their names in a tattered notebook held together by duct tape and managed by the migrants in a plaza outside the entry to the main border crossing.

On Thursday, migrants who registered six weeks ago were getting their names called. The waiting list has grown to more than 3,000 names and stands to become much longer with the caravans.

Francisco Rueda, the top deputy to Baja California state Gov. Francisco Vega de la Madrid, said that if all migrants from the caravan currently in Tijuana were to register to seek asylum in the U.S., they would likely have to wait four months in Mexico at current processing rates. For that reason, the state has asked Mexican federal authorities to encourage others in caravans to go to other border cities.

Rueda reported that about 1,750 migrants from the caravan had now reached Tijuana. Private shelters can house about 700, and the city-owned gym and sports complex has a capacity of 1,000 with the potential to expand to 3,000.

“This is not a crisis,” he told reporters.