They planned to apply for asylum at the American border, but knew there was a good chance that they would be split up during the process — possibly for months.
“But I’m going with the feeling that it’s going to be worth the effort,” said Mr. Quintanillo. He said his family were fleeing a gang that had attacked him and killed a close relative. “In the name of God, everything is possible,” he said.
Overlaying the personal struggles was a dense tangle of politics and policy — the ill will between Mr. Trump and Mexico that began the day he announced his candidacy; the acrimony between Mr. Trump and Gov. Jerry Brown of California over immigration; the politics of sanctuary cities; and the political logjam in Congress over funding Mr. Trump’s proposed border wall.
It all plays out in the context of Mr. Trump’s goal of making immigration a galvanizing issue in the midterm elections with Republicans worried about losing control of the House and perhaps the Senate.
Heather Cronk, co-director of Showing Up for Racial Justice, one of several American advocacy groups that have been helping the caravan and its participants, traveled to Tijuana to support the migrants in the final stretch.
“For us, this is all about who we are as a country,” she said. She added: “This is an existential moment. This is a spiritual moment. I want it to be true that when we say, ‘Liberty and justice for all,’ we mean it.”
It is a debate Mr. Trump apparently relishes.
With the migrants on the doorstep of the United States, Mr. Trump, in a tweet last week, ratcheted up his rhetoric, vowing “not to let these large Caravans of people into our Country.”
Mr. Trump repeatedly came back to immigration issues at a rally in Michigan on Saturday night, saying at one point: “If we don’t get border security, we’ll close down the country,” apparently referring to a government shutdown when a funding deadline is reached in September.
Other administration officials have also been vocal.
Attorney General Jeff Sessions called the caravan “a deliberate attempt to undermine our laws and overwhelm our system.”
Joined by supporters and dozens of members of the news media, the migrants gathered in a park on the Pacific Ocean about 10 a.m. local time and then later on a pedestrian plaza in front of a community center in downtown Tijuana. Scores of supporters, some of whom had walked from as far as Los Angeles, rallied Sunday morning just north of the fence separating the United States from Mexico on the American side of the oceanfront park.
What was supposed to be the final act of the caravan began about 3:30 p.m., when more than 150 of the participants, accompanied by relatives, supporters and the press, marched several blocks to a border crossing in Tijuana called El Chaparral. As they walked, they chanted and waved Honduran flags.
To qualify for asylum, applicants must prove they have been persecuted or fear persecution based on their race, religion, nationality, political beliefs or membership in a particular group.
People who request protection at a United States entry point must be referred to an asylum officer for a screening, known as a credible-fear interview. If the officer finds that an applicant has a chance of proving fear of persecution, the person must then present his or her case before a judge. More than three quarters of applicants pass that initial review.
“We’re only sending people who we think will pass the credible-fear interview,” said Nicole Ramos, a volunteer immigration lawyer helping the caravan.
But Customs and Border Protection, whose officers are stationed at ports of entry, announced late Sunday that it had exhausted its capacity to handle people traveling without documents.
Still, caravan organizers escorted some 50 participants along the long, elevated pedestrian walkway at El Chaparral that leads from Tijuana to the entrance to the United States in San Diego. At the gate leading into the American immigration checkpoint, American border authorities reaffirmed that they would not be able to process any more asylum-seekers on Sunday.
Alex Mensing, project coordinator for Pueblo Sin Fronteras, a transnational group that organized the caravan, told reporters gathered at El Chaparral that the migrants would remain at the gate, overnight if necessary, until border officials once again had the capacity to process them.
“We wish that the United States government were capable of accepting more than a few hundred asylum seekers at any given time, since we can certainly pick up more than a 1,000 people in an ICE raid on any given day,” he said, referring to Immigration and Customs Enforcement, an arm of the Homeland Security Department.
Meanwhile, the rest of the asylum seekers, their relatives and supporters laid out blankets on a plaza outside the entrance to El Chaparral and prepared for a long, chilly night.
When they get a chance to make their case, migrant families that request asylum at the port of entry are likely to be placed on buses to Texas, where they will remain in detention centers for mothers and children. Adult men are likely to be detained in any number of facilities across the country that hold undocumented immigrants.
It is in these facilities that the migrants would be screened by United States immigration officials over the next several days. If they pass the credible-fear interview, the migrants will be allowed to make their case for asylum before an immigration judge, a process that unfolds over several months or longer.
Migrants, typically fitted with ankle monitors, often are allowed to travel to the interior of the country, where they stay with relatives or friends while their cases run their course.
Mr. Trump, however, has denounced that practice because some migrants have skipped their court hearings; he dismissed it as “catch and release.” In recent months, migrant advocates say, the Trump administration has kept many migrants seeking asylum in detention.
For all the high political stakes, the human stakes for the individual migrants planning to seek asylum Sunday were at least as high.
Byron Claros, a Salvadoran immigrant, joined the caravan with his 18-year-old brother, Luis Alexander Rodriguez, and their stepfather, Andres Rodríguez.
Mr. Claros and Mr. Rodriguez planned to petition for asylum Sunday afternoon; their stepfather, after consultation with volunteer lawyers in Tijuana, decided that his case for sanctuary was not strong enough and that he would remain behind in Mexico.
“The hour I’ve waited for my entire life has finally arrived,” Mr. Claros said early Sunday afternoon as he, hundreds of migrants, scores of their supporters, reporters and cameramen gathered in and in front of a community center and cafe in the downtown district of Tijuana, blocks from the border crossing.
Mr. Rodriguez said he was nervous, “because the United States can support our rights but can also deny us our rights.”
Still, he said, there was only one way to push: north.
“We’ve fought too much to get here,” he said. “And we’re here.”