MAASTRICHT, Netherlands—The picturesque town that gave its name to the pact that created the modern-day European Union is now a bastion of euroskeptics who want to follow Britain out of the bloc.

Town resident Armand Vliegen supported the drive toward a more integrated Europe when the Maastricht Treaty came into force in the early 1990s. But when dozens of asylum seekers recently settled into a converted student dorm about 20 yards from his home, he said his little remaining pro-EU sentiment evaporated.

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