Eighteen-year-old Alfonso Lucio remembers the sun beating down on the back of his neck as he toiled in asparagus fields in Michigan, something he began doing at age 12. His cousin Jazmine Hernandez, who is the same age and also came from a family of migrant farm workers, was often at his side.

“Just wait,” he would tell her. “A few more years of this, and then we’re going to college.”

It took the help of a decades-old federal program that focuses on migrant students, but he was right. Last month, as the cousins had lunch in a dining ...

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