Driven by a deep faith rooted in Christianity, many have clung to a belief that everything would work out in the end, that Mr. Trump’s heart would be touched and he would allow them into the United States to work.

Even Ms. Alvarado subscribed to this hopeful philosophy.

“Well, I can’t think the opposite way,” she shrugged.

While most migrants in the caravan are young men, the group also includes many families with young children who have now been traveling for four weeks.

Ms. Alvarado and her relatives left their home on the outskirts of Comayagua, a city in central Honduras, on Oct. 12. They came from a family of farm laborers who worked for abysmal wages in coffee plantations. Generations of residents from Comayagua had made the trek to the United States to find better-paying work, and the possibility was always forefront in the minds of those who remained behind.

Ms. Alvarado, one of the few in her family who had managed to escape the coffee fields, had been working as an assistant in a government social development program, but barely getting by on a salary of $200 a month.

In October, she heard on television about the caravan taking shape in San Pedro Sula and decided it was her best chance to make it to the United States’ southern border. Within a couple of days, Ms. Alvarado had set off to join it. The other two women and their children went, too, viewing the caravan as their safest and least expensive opportunity to migrate.

Each mother had a different destination in the United States in mind; the group’s relationship with the country was already complex.