SANTIAGO, Chile—Every night, a commercial airliner full of Haitians arrives in this city’s international airport carrying families who have bid adieu to their poor island nation for a new life here.

As the Trump administration aims to curb immigration, one of Latin America’s richest and safest countries has opened its doors to some of the region’s poorest migrants in record numbers. Tens of thousands of Venezuelans have fled their crime-ridden country in recent years for Chile, which has a history of receiving Bolivian, Peruvian...