Joseph, who was hired Tuesday, has been a defensive backs coach in the NFL since 2006. He was with the 49ers for five years and the Texans for three before joining the Bengals in 2014. Joseph played in the NFL in 1995-96.
The Bengals had the fifth-best opposing quarterback rating in the league this season, and ranked third with 21 interceptions.
Miami defensive co-ordinator Kevin Coyle was fired in October after 3 1/2 seasons and replaced by Lou Anarumo. The Dolphins ranked 22nd in points allowed, gave up a franchise-record 4,000 yards passing and finished 6-10.
Gase, a former offensive co-ordinator for the Bears and Broncos, was hired Saturday.