TIJUANA, Mexico — Several members of the Latin American migrant caravan that has enraged President Trump were allowed to step onto United States territory to apply for asylum late Monday, ending a border standoff that had lasted more than a day and marking the beginning of the final chapter of the group’s monthlong odyssey.

Shortly after 7 p.m. local time, eight migrants who, like most of the caravan’s participants, said they were fleeing violence in their homeland, passed through the metal gate separating Tijuana from San Diego, entered the immigration checkpoint and began the process to petition for sanctuary, caravan organizers said.

The contingent that was admitted included four children and three women — the children’s mothers — and an 18-year-old man. The organizers said they did not know whether more of the migrants would be permitted to enter Monday night.

The news was greeted with a guarded sense of relief by the caravan’s participants, many of whom had been hunkering down for more than a day in a makeshift encampment at the entrance to the border crossing waiting for the logjam to break.