TIJUANA, Mexico — Exhausted migrants in a caravan of Central American asylum-seekers napped on mattresses in a converted municipal gymnasium, while men played soccer and exchanged banter on a crowded, adjoining courtyard. A woman dabbed her crying, naked toddler with a moist cloth.

Nearly 2,000 caravan migrants had reached the U.S. border in Mexico’s northwestern corner by Thursday, with more coming in a steady trickle of buses. The city of Tijuana, with its privately run shelters operating well above their capacity of 700, opened the gymnasium and gated sport complex for up to 1,000 migrants, with a potential to expand to 3,000.

With U.S. border inspectors processing only about 100 asylum claims a day at the main border crossing with San Diego, prospects grew that migrants would be stuck waiting in Tijuana for months.

Francisco Rueda, the top deputy to Baja California state Gov. Francisco Vega de la Madrid, said about 1,750 migrants from the caravan had reached Tijuana so far.

“This is not a crisis,” he told reporters, though he agreed that “this is an extraordinary situation.”

Rueda said the state has 7,000 jobs available for its “Central American migrant brothers” who obtain legal residence status in Mexico.

“Today in Baja California there is an employment opportunity for those who request it, but it order for this to happen, it has to regulate migrant status,” he said.

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 in Tijuana the last two years.

Police made their presence known in a city that is suffering an all-time-high homicide rate. A group of about 50 migrants, mostly women and children, walked through downtown streets Thursday from the city shelter to a breakfast hall under police escort.

As buses from western and central Mexico trickled in, some families camped inside the bus terminal and waited for word on where they could find a safe place to sleep.

Oscar Zapata, 31, reached the Tijuana bus station at 2 a.m. Thursday from Guadalajara with his wife and their three children, ages 4, 5 and 12, and headed to the breakfast hall, where migrants were served free beef and potatoes.

Back home in La Ceiba, Honduras, he sold pirated CDs and DVDs in the street and two gangs demanding “protection” money threatened to kidnap his daughter and force her into prostitution if he didn’t pay. When he heard about the caravan on the TV news last month, he didn’t think twice.

“It was the opportunity to get out,” Zapata said.

Zapata said he hopes to join a brother in Los Angeles but has not yet decided on his next move. Like many others, he plans to wait in Tijuana for others in the caravan to arrive and gather more information before seeking asylum in the United States.

Byron Jose Blandino, a 27-year-old bricklayer from Nicaragua who slept in the converted gymnasium, said he wanted to request asylum but not until he could speak with someone well-versed in U.S. law and asylum procedures.

“The first thing is to wait,” Blandino said. “I do not want to break the laws of any country. If I could enter in a peaceful manner, that would be good.

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 new arrivals.

Dozens of gay and transgender migrants in the caravan were already lining up Thursday to submit asylum claims, though it was unclear how soon they would be able to do so.

Rueda, the governor’s deputy, 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 at current processing rates. For that reason, the state has asked Mexican federal authorities to encourage people in other caravans to go to other border cities.

There are real questions about how the city of more than 1.6 million will manage to handle the migrant caravans working their way through Mexico, which may total 10,000 people in all.

“No city in the world is prepared to receive this number of migrants,” said Mario Osuna, Tijuana’s social development director. He said the city hopes Mexico’s federal government “will start legalizing these people immediately” so they can get jobs and earn a living.

The caravan has fragmented somewhat in recent days in a final push to the border, with some migrants moving rapidly in buses and others falling behind.

On Thursday, hundreds were stranded for most of the day at a gas station in Navojoa, some 750 miles (1,200 kilometres) from Tijuana.

“We were dropped here at midnight … in the middle of nowhere, where supposedly some buses were going to come pick us up, but nothing,” Alejandra Grisel Rodriguez of Honduras told The Associated Press by phone. “We are without water, without food.”

After about 12 hours, seven buses began arriving to collect the migrants, Rodriguez said, but they quickly filled up.

“We would need at least 40 or 50,” she said.

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Associated Press writer Maria Verza in Culiacan, Mexico, contributed to this report.

Elliot Spagat, The Associated Press