DOVER, England — Trucks rumble nonstop through Britain’s busiest ferry port, one every few minutes, some headed for France, others arriving here in Dover, where they trundle along an elevated roadway that scales the town’s striking white cliffs.

But a cloud is hanging over the ceaseless flow of cargo to and from continental Europe — and it is more than a fog blown in from the English Channel.

Of the thousands of trucks that use Dover each day, only a fraction are stopped by British officials. That is because both Britain and France are members of the European Union and of its customs union, which removes the need for border checks on most goods.

Yet when Britain quits the bloc, a process known as Brexit, all that could change, and the question of how, when — or whether — to abandon a European customs union has divided and paralyzed the British cabinet.