One of the oldest names on Wall Street is moving to one of the fastest-growing cities in the South, reinforcing a recent shift in finance jobs to cheaper parts of the U.S.

AllianceBernstein Holding LP plans to relocate its headquarters, chief executive and most of its New York staff to Nashville, Tenn., in an attempt to cut costs, according to people familiar with the matter. That largely ends a 51-year presence in the nation’s traditional finance capital.

The...