The city, which spends $500,000 a year on the program, employs a local travel agency, Austin Travel, to book one-way tickets for domestic trips. Department of Homeless Services employees do all the planning for international travel.

City officials said there were no limits on where a family can be sent, and families can reject the offer and stay in city shelters. So far, families have been sent to 24 states and 5 continents, most often to Puerto Rico, Florida, Georgia and the Carolinas.

“We want to divert as many families as we can that need assistance,” said Vida Chavez-Downes, the director of the Resource Room, a city office with 11 social workers, two managers and an administrative assistant who help relocate families. “We have paid for visas, we’ve gone down to the consulate, we’ve provided letters, we’ve paid for passports for people to go. Anyone who comes through our door.”

One family with 10 children accepted an offer to go to Puerto Rico on a nonstop JetBlue flight. An adventurous but ultimately unlucky Michigan couple drove to the city in search of jobs and a new life. They got $400 in gas cards to drive back.

One set of parents agreed to move to France with their three children to be with the mother’s family. The $6,332 travel cost included five plane tickets to Paris and five train tickets to the town of Granville, in the northwest.

In the past, the city contracted with the Salvation Army for a now-defunct program called Homeward Bound, but only for single adults and couples, not families with children. Both versions followed the example of Travelers Aid, a 150-year-old nonprofit organization that provides stranded and homeless people emergency aid so they could return to their homes, and which still exists today. Other cities have experimented with similar programs, but they are largely focused on adults without children.