MOSCOW—Sometimes the wheels of history turn slowly. The hottest shopping destination for Cubans is not across the water in Miami. It’s Moscow, 6,000 miles away.

Tougher U.S. border control and rising remittance income from relatives abroad have led to a recent surge of Cuban travel to Russia, the only major country that still doesn’t ask islanders for a visa. Cuban shoppers don’t take the daily 13-hour Aeroflot flight, a legacy of the Soviet-era alliance, to see the Kremlin or the Red Square. They bring back bags of jeans,...