More than a year before Britain is scheduled to depart from the European Union, signs are emerging that the largest and most sustained influx of migrants in the country’s modern history is coming to end.

For many backers of Brexit, this is welcome proof that the decision to leave the bloc was right. Over the last 20 years, Britain experienced an unprecedented rise in immigration, particularly from new, poorer members of the EU. That influx, blamed for putting pressure on local health services, housing and education, was a...