A migrant train of around 3,000 people continued their trek through the Mexican state of Chiapas on Friday.
Many had come from Central America and the Caribbean, and had started walking last weekend from the southern border to reach the U.S. or Mexico City.
As they arrived in the town of Acacoyagua, Mexican authorities offered them visas that would let them provisionally settle in Mexico, in exchange for ending their journey.
One of the leaders said one of the government's conditions was to dissolve the caravan and transport migrants on buses to different parts of Mexico.
It was put to a vote, and many declined the offer, including Salvadorian migrant Rosa Elena.
"We think it's a trap. It's hard for us to accept this opportunity Mexican authorities are offering because they are saying they want to separate us, which will make it easier for them to capture us."
The humanitarian visas would allow the migrants access to healthcare, as well as the right to work.
But many are distrustful of migration officials, due to what's described as broken promises as well as arrests and deportations.
The Mexican government is under pressure from Washington to stem the flow of migrants from the south.
And Mexican authorities have obliged by beefing up patrols.