At the December day’s first glimmer of light, a small Honduran woman carried her feverish 3-year-old daughter along the empty streets of East Baltimore toward Johns Hopkins Hospital, a place she had never heard of but could just see in the distance.

Josselin Mencia came to the United States because she had no faith that Honduran doctors could help her daughter, a tiny, fragile child who was born partially paralyzed. They were among 60,000 people who were detained at the southwestern border in December, roughly half of them families with children.

Mencia and her daughter, Brianna, have no certain future, though there are possible legal routes for seeking asylum when someone needs medical treatment, lawyers say.