LIMA, Peru—It was supposed to have been what the American family called “a bucket list” trip for Lesley Bravi, 77, whose husband had died last year: A trip to Peru with relatives to see the 15th-century Inca citadel of Machu Picchu and Lake Titicaca.

But on Monday, a week into her trip, an email came from the U.S. Embassy in Lima, telling Ms. Bravi and her daughter, Kathleen de Chabert-Ostland, that they had to get out of Peru fast. With the coronavirus pandemic on the march, Peru had declared a state of emergency, enacted...