MIAMI - Uruguay’s banking crisis had already claimed Rosana Araujo’s savings when she and her family, tourist visas in hand, boarded a Miami-bound plane in Montevideo. She stayed in Miami after her visa expired, supporting herself on random jobs, cleaning homes or babysitting. “I feel lucky because I got here in a plane,” she said. “I feel lucky because I didn’t have to cross a desert.” While the national discussion on immigration focuses on securing the U.S.-Mexico border with a multibillion dollar wall, travelers such as Araujo who overstay their visits have become the main source of illegal immigration in...