The married couples filed into a federal building in San Diego last week for green card interviews that they believed would secure their future together in the United States. Half of each pair was American. Stephen Paul came with his British wife and their 4-month-old baby. Audrey Hestmark arrived with her German husband, days before their first wedding anniversary. Jason Cordero accompanied his Mexican wife.

It was supposed to be a celebratory milestone, the final step in the process to obtain U.S. permanent residency. Instead, as each interview with an immigration officer wrapped up, federal agents swooped in, handcuffed the foreign spouse and took him or her away.

“I had to take our baby from my crying wife’s arms,” Mr. Paul, 33, said, recalling the moment that agents said they were arresting his wife, Katie.

Ms. Paul was sent to an immigration detention center with hundreds of other people swept up in the Trump administration’s crackdown. Her husband had to take a leave from his job at the San Diego County Sheriff’s Department to care for their child and try to secure her release.