On Oct. 4, Jason Dubin drove to La Guardia Airport to pick up the newest member of his household. Six days after that first happy meeting, Mr. Dubin returned to La Guardia with that same member, a German shepherd named Emmi, and sent her on a Continental Airlines flight to Seattle with a one-way ticket. “I just couldn’t control her anymore; it was just time to part ways,” said Mr. Dubin, who made the drive from his Port Jefferson home on Long Island as Emmi, who is 5, fidgeted and barked in the back seat. Mr. Dubin had bought the...