When unaccompanied children are apprehended at the border — often after having been taken there by smugglers — immigration officials initiate cases for their deportation, a process that can take months or years. In the meantime, many of those children are placed with parents or relatives who crossed earlier to establish a foothold in the United States and earn money to send back home.

Until recently, those adults have not been priorities for arrest, even if they are in the country illegally.

But in February, President Trump’s Homeland Security secretary, John F. Kelly, signed a memo promising to penalize people who pay smugglers to bring their children to the United States, saying that the agency had “an obligation to ensure that those who conspire to violate our immigration laws do not do so with impunity.” This past week, Jennifer D. Elzea, the deputy press secretary for Immigration and Customs Enforcement, confirmed that arrests had begun.