TSAREVO, Bulgaria—Police this month chased a late-model Toyota Rav4 up the Black Sea coast from Turkey, stopping it at a roadblock. Inside, they found the rear seats had been removed to make room for 11 Iraqi migrants, including a mother and her child. The driver, it emerged, was a local convict out of jail on a medical furlough.

Bulgarian police have hauled hundreds of smugglers, drivers and even fellow officers into court in recent months as the European Union locks down some of the weak spots on its periphery against migrants....