Pueblo Sin Fronteras, a group of activists escorting the caravan, warned the migrants that the offer might be too good to be true and called a voice vote on whether to continue.
"Let's keep going!" the crowd yelled amid applause.
And they kept going. Thousands are now in Tijuana on the U.S. border, where they are likely to be camped for months or longer with no easy way to get into the United States, creating what is fast becoming a humanitarian crisis in this overwhelmed city.
Many blame Pueblo Sin Fronteras, or People Without Borders, made up of about 40 U.S. and Mexican activists.
Critics, including former allies and some of the migrants themselves, say Pueblo Sin Fronteras downplayed the dangers of such treks, especially for families and small children, and misled the participants about how long they would have to wait on the Mexican side to apply for asylum.
Adelaida Gonzalez, 37, of Guatemala City, who joined the caravan with her 15-year-old son and neighbor, said that now that she is in Tijuana, she wishes she had accepted Mexico's offer to stay and work in the southern Mexican state of Chiapas.
"We were never told along the way that it would be this hard," said Gonzalez, after seeing the border wall topped with razor wire and the long waiting list for asylum seekers.
A Pueblo Sin Fronteras leader, Irineo Mujica, emphatically rejected the criticism.
"Our commitment first and foremost was protecting the lives of migrants and giving them as much information as possible," Mujica said. "To blame the people who are helping is crazy."
Pueblo Sin Fronteras founder Roberto Corona said in the organization's defense that attorneys along the way told the migrants they could be held in U.S. detention centers for months and possibly separated from their children. In the end, he said, the migrants — many of whom are fleeing poverty and violence in Honduras, El Salvador and Guatemala — made their own decision.
"They know the wall is very big, and they will not be very welcome in the U.S. by many people," Corona said, "but still they have hope of coming here, that at least their rights will be more protected, and they will be able to make a living."
This is the fourth and biggest caravan of Central American asylum seekers that Pueblo Sin Fronteras has helped reach Tijuana, a trek that angered President Donald Trump and prompted him to send troops to the border. When the caravan crossed into Mexico, it numbered 7,000; about 5,500 made it to Tijuana.
Pueblo Sin Fronteras maintains it simply accompanies the migrants to protect their rights. But the organization clearly plays an essential role: It helped charter the route, arrange bus transportation and negotiate with Mexican officials to provide protection. It also raised more than $46,000 online for emergency housing and food.
As the caravan crossed Mexico, the organization held nightly assemblies to decide the next day's destination. It alerted towns to prepare for migrants who camped in their squares.
For the Central Americans, there was a feeling of safety in numbers. For decades, migrants crossing Mexico have been robbed, kidnapped and killed by gangs and corrupt officials.
But traveling with the caravan was not without risks. One migrant was killed when he fell off a truck. Another was run over and killed on a highway. Two were stabbed and strangled after leaving a Tijuana shelter. Others have been attacked with rocks by local residents angry over the mass arrival.
Last month, a march by the migrants in Tijuana to demand the United States accelerate its asylum process degenerated into violence. Demonstrators threw rocks at U.S. border agents and tore down fencing, letting dozens rush through. U.S. authorities fired tear gas into Mexico, sending crowds that included children running and screaming.
"There is no reason to make these inhumane journeys," Alejandro Solalinde, a Mexican priest recognized for his work with migrants, said of the caravans.
Sergio Tamai, whose organization operates migrant shelters in the Tijuana area, said he called Mujica to express concerns about the Tijuana march beforehand — "and we all saw what happened — a disaster."
Migrants say they are grateful for all Pueblo Sin Fronteras has done, but they were not prepared for the long wait in Tijuana. Some 3,000 people were already in line to ask for asylum before the caravan arrived, and U.S. authorities are processing only about 100 claims per day at the crossing, resulting in overflowing shelters in Tijuana.
And the wait just got longer: The Trump administration announced Thursday that asylum seekers at the border will now be forced to wait in Mexico while their cases slowly wind their way through the clogged U.S. immigration courts. Previously, migrants were allowed into the U.S. while their claims were processed.
Esmeralda Siu, a Tijuana shelter manager, said many caravan members knew nothing about the difficulty in getting asylum.
"They come in desperation and so they hear what they want to hear," she said. As for their escorts, "it seems like they are putting the migrants at great risk."
Corona, Pueblo Sin Fronteras' founder, said that the caravans served their purpose but that he doesn't foresee the organization accompanying any more of them.
"We need to find out how can we reach the hearts and minds of the American people and Mexican people and mostly the policymakers who can come up with a permanent solution," he said.
Associated Press writer Christopher Sherman in Mexico City and researcher Jennifer Farrar in New York contributed to this report.