Usually a professional franchise of any kind will only relocate to a new city because of a few factors. Poor stadium, lack of fan interest, better offer by a new city, are all situations in which pro teams have chose to relocate their franchise. Two NFL moves stand out in history, more for their timing regarding team performance than anything else.
The Kansas City Chiefs franchise won their first championship while members of the American Football League in 1962. At the time of the victory, the team played in Dallas and went by the name Dallas Texans. The Texans were a success; they had won a league championship in only their third year of existence. The very next year, 1963, the Texans moved to Kansas City and became the Chiefs. The move was as a result of the popularity of the relatively young NFL franchise Dallas Cowboys.
In 1945, the Rams had won the NFL Championship and were on top of the world. That following season found the Rams playing under a new name; the Los Angeles Rams. Having won the title as the Cleveland Rams, from their time spent sharing Cleveland with the Browns, many fans couldn’t watch their team try to defend their title after they had moved out to California.
A team moving to new cities does happen occasionally; the Tennessee Titans were once the Houston Oilers, the Indianapolis Colts were once the Baltimore Colts, and the Arizona Cardinals were once the St. Louis Cardinals (and the Chicago Cardinals before that), but few teams move the very next season after winning the Championship.