WASHINGTON—The Trump administration moved to allow the government to indefinitely detain families crossing the U.S.-Mexico border and supersede a decades-old court settlement that both limits how long migrant children can be held in custody and sets standards for their care.

The new rules are the Republican administration’s latest effort to tighten immigration laws on its own, with Congress long unable to agree on any legal overhaul. Wednesday’s policy change could permit authorities to detain families through the duration...