Adam and Amanda Powell with their daughters, Charlotte and Olivia, outside of Dallas, Texas, where the family looked for a new home. Shelby Tauber for The New York Times

Adam and Amanda Powell loved living in Liechtenstein, Europe’s fourth-smallest country. The American couple, who work for Hilti, a Liechtenstein-based power-tool company, spent three years in a modern, four-bedroom apartment in the city of Schaan with their two young daughters. But last winter, they were told that their work visas could not be renewed and that they were being relocated to Hilti's North American headquarters just outside of Dallas.