TIJUANA, Mexico — A standoff between a caravan of Latin American migrants and the United States government continued for a second day here on Monday as American border authorities declined to process asylum seekers. Migrants hunkered down in a makeshift encampment at the entrance to a border crossing in Tijuana, and refused to budge.

The stalemate began on Sunday afternoon when Customs and Border Protection, the arm of the Department of Homeland Security that regulates ports of entry, announced that it had no more capacity to process asylum-seekers at its San Ysidro border entry in San Diego, which abuts Tijuana.

Still, about 200 migrants from the caravan who intended to seek asylum — accompanied by other caravan participants, supporters and dozens of journalists — set off in a procession from downtown Tijuana to the El Chaparral border crossing, setting up the standoff.

Denied entry to United States to make their case for sanctuary, the migrants, nearly all of them from Central America, settled down in the pedestrian plaza in front of the Mexican border entrance.

As the cold night fell, they unpacked the few warm clothes they carried in knapsacks and plastic bags, huddled under donated blankets and ate sandwiches distributed by volunteers. By daybreak, they had strung up plastic tarps to shield them from the sun and rain, and vowed to stay until the asylum seekers among them were permitted to step on American soil and petition for protection.

Customs and Border Protection officials did not respond to requests for comment on Monday, and the Mexican authorities referred inquiries to their American counterparts, saying it was not a Mexican administrative matter.

The caravan’s organizers said that the migrants were a tough group and that the wait, however uncomfortable, was a minor challenge compared with the hardships they had already faced — in their homelands, where they faced violence and poverty, and during their strenuous trip north.

“This is the least of their suffering,” said Irineo Mujica, a member of Pueblo Sin Fronteras, the transnational advocacy group that organized the caravan.

The caravan began on March 25 in Tapachula, a city on Mexico’s southern border with Guatemala, and it quickly grew to include more than 1,200 migrants. Such mass migrations, which have become something of an annual tradition in Mexico, provide migrants with safety in numbers while giving advocates a platform to air their criticisms of regional migration policies.

This caravan, in part because of its extraordinary size, came to the attention of President Trump, who in a barrage of messages on Twitter said that the caravan threatened the security and sovereignty of the United States. He also accused Mexico of doing little to stop illegal migration.

Undeterred, the caravan continued north, its participants traveling atop freight trains and by hitchhiking, sleeping in gritty migrant shelters and parks and eating poorly. Hundreds dropped out along the way, choosing to travel on their own or remain in Mexico.

More than 300 participants arrived in Tijuana last week and were joined in recent days by supporters from Mexico and the United States, including scores of volunteer lawyers and paralegals who have converged on this border city to provide free consultations to the caravan’s participants.

The organizers and lawyers identified the strongest asylum cases — about 200, most of them children — and encouraged those people to apply for protection, while suggesting that the remainder seek protection in Mexico or elsewhere in Latin America.

To qualify for asylum, applicants must prove that they have been persecuted or fear persecution based on their race, religion, nationality, political belief or membership in a particular group. People who request protection at an American entry point must submit to a screening by an asylum officer, called a credible-fear interview. If the officer finds the fear credible, the case is then referred to an immigration judge for a full hearing.

But since arriving at the border on Sunday, none of the asylum seekers among the caravan have been allowed by border officials to begin the process.

Migrants’ advocates accused the Trump administration of engaging in brinkmanship for political gain at the asylum seekers’ expense.

“Customs and Border Protection is the largest law enforcement agency in the country, and is able to detain, transport and incarcerate thousands of people in a day, but is pretending that they don’t have the ‘capacity’ to accept 150 refugee parents and children whose arrival has been anticipated and communicated weeks in advance,” Pueblo Sin Fronteras said in a statement on Monday.

In the vacuum of authoritative information, rumors circulated among the migrants and their supporters.

Caravan organizers said they had been told by Customs and Border Protection agents that no asylum seekers had been processed at the San Ysidro entry for three days, and that the caravan’s migrants could be waiting another several days or longer before they would get a chance to petition for asylum.

Most of the migrants, however, had adopted mind-sets of hardened patience.

“Really, nobody’s said anything.” said Arnaldo Rivera, 40, who fled his native Honduras with his wife and five children after the family was threatened by a gang. They were among about 200 migrants who were waiting outside the entrance to the border crossing on Sunday, eating donated food and using a nearby public bathroom.

The family had staked out a patch of the pedestrian plaza by spreading out a blanket and demarcating it with a few knapsacks containing their worldly belongings.

“It could be this afternoon, it could be tomorrow,” he said, shrugging. “God has the last word.”

Late Sunday, local, state and federal authorities tried to persuade the migrants to decamp from the pedestrian plaza and spend the night in migrant shelters. But in an act of communal defiance, the caravan’s participants elected to remain where they were. As the officials walked away, the migrants applauded and cheered.

“We’ve experienced a lot and this isn’t going to stop us,” said Shannel Smith, 29, who fled gang violence in Honduras and is one of about 35 transgender migrants in the caravan.

Ms. Smith was part of a delegation of about 50 migrants who were selected by the lawyers and caravan organizers to approach the American border entry on Sunday afternoon and test the authorities’ claim that they had no capacity to process asylum petitions.

But the members of the group, many of them children, were not allowed to pass through the gate. Some decided to remain there for much of the night, corralled behind metal barricades and huddled under blankets on the cold concrete, maintaining pressure on the American government.