If a professional team franchise moves to a new city do the records for that team remain with the city or with the franchise?

The team regardless of any name or city changes is the franchise. The records of that franchise will always follow the team. The City that the franchise played in could only make claim to team records whiles that team was playing for that city.

For Example:

The Atlanta Braves franchise started in Boston in 1871, and went though many name changes. But it is still the same franchise. The team moved to Milwaukee in 1953, and then to Atlanta in 1966. Different cities, but still the same franchise, or team.

The Atlanta Braves franchise won three World Series in three different cities. Boston, Milwaukee, and Atlanta. The Atlanta Braves franchise could hold claim to all three World Series titles but the city of Boston has no claim to the titles won in Milwaukee and Atlanta. (I think that came out right)

Think of the team as a company. If a business moves from one city to another it is still the same company. It just conducts business in a different location.

One time that this did not happen is with the Cleveland Browns football team in 1996. The owner of the Browns announced the team was moving to Baltimore, where it ultimately became the Baltimore Ravens. All players who were on the Browns roster in 1995 went to Baltimore. The Browns, as a team, were deactivated for 3 years then reactivated in 1999 and restocked with players by way of a special draft. Normally, all the past team records of the Browns would have gone with the team to Baltimore, but in this case, the new Cleveland team retained both the team name and all of its records. All of the league championships the Browns won stayed in Cleveland. In contrast, when the Baltimore Colts went to Indianapolis, they took with them all of their previous history.

The Baltimore Ravens started out designated as an expansion team with no history behind it despite the fact that it was an existing team that simply moved to another city. The current Cleveland Browns appear to be a team with championship calibre history behind it despite the fact that the team itself is in reality an expansion team of new players.