The Braves had moved from Boston to Milwaukee for the '53 season, and that marked the first National League change since the league shrank from twelve teams to eight for the 1900 season. The Braves' move sparked the flurry of migrations that soon ensued.
The Browns' move in 1954 made them the first American League team to uproot to another city since 1903, when the original Baltimore Orioles moved to NYC and became the Yankees.
They were followed by the Philadelphia Athletics' 1955 move to Kansas City (where after last night they probably wish they had remained instead of moving on later to Oakland), the Brooklyn Dodgers & NY Giants moving to Los Angeles and San Francisco in 1958, and the Washington Senators' move to Minneapolis in 1961.
To complete the study: the Milwaukee Braves became the Atlantas, the Seattle Pilots became the Milwaukee Brewers, the second Washington Senators team became the Texas Rangers, and the Montreal Expos became the Washington Natinals. I'm unaware of any other relocations (leaving aside expansion teams and the Astros' recent league-change), although the Angels are a curious case: born in Los Angeles, then moved 35 miles to Anaheim, then renamed the L.A. Angels while remaining in Anaheim. But then, I've always been a Dodger fan.
According to Wiki the Milwaukee Brewers of the Western League, which became the American League, were moved in 1902 to St. Louis to become the St. Louis Browns. I did not realize there was a Brewers team in Milwaukee before the current team.