I'm not too familiar with NHL, MLB or the NFL. I do know that a number of NBA franchises have a few moves in their history, although some of the moves predated the actual league itself.
The name that has bounced around the most has probably been the Hornets. They started as a late '80s franchise in Charlotte, before the franchise moved to New Orleans after the turn of the century. That was the first move. Hurricane Katrina wound up meaning the team played in Oklahoma City for a season, then went back to New Orleans for the second and third moves. Eventually, the New Orleans franchise and ownership wanted a Louisiana-specific name, and the new Charlotte Bobcats wanted the Hornets name back, so the name jumped cities and franchises at the same time in 2014!
As far as the same team to make moves, I think in basketball, that honor belongs to the Sacramento Kings, which moved there from Kansas City in 1985. They had played there since 1975, and split time between two cities as the Kansas City-Omaha Kings from 1972-1975. Prior to that, they were the Cincinnati Royals from 1957 on, and the Rochester Royals before that.