First of all, here are the champions of each league throughout the years.
Lots of surprising dynasties. The Senators won five straight pennants from 1909-1913. The Phillies took nine straight pennants 1925-1933. The Tigers dominated with five straight 1947-1951. The Yankees don't have their first long string of pennants until 1971-1974. The Giants took six in seven years 1986-1992, and faced the White Sox four straight years 1988-1991. The Expos finally got their pennant in 1996. It did not keep them in Montreal.
One kind of shocker: the Dodgers won four World Series in five years 1953-1957, yes still moved to LA for 1958 (where they won another pennant). I gotta think that if they'd won four Series in those five years, they would have stayed in Brooklyn, and some other team would have made their way out to LA, like the Cubs.