I didn't have Civ4, could you please explain what you mean by that?
Very roughly, each tile within your cities range received an amount of cultural pressure per turn, based on how much culture your city is generating per turn, distance and maybe age (how long you have been applying cultural pressure on said tile for). If your civ had the most cultural pressure on a tile - you got to control that tile.
Example - City of Oldham and Newfax. For simplicity the tile the two cities are disputing is 2 tiles away from both cities.
Oldham generates +10 culture per turn
Newfax generates +5 culture per turn
Result = Oldham controls the tile. 66% of the population of that tile is considered to belong to Oldham (10 / (10 + 5)) while the remainder is Newfaxian.
Later, Newfax builds a wonder that generates +10 culture per turn
Oldham generates +10 culture per turn
Newfax generates +15 (5 + 10) culture per turn
Result = Oldham controls the tile initially, but the tile population gradually becomes Newfaxian. 20 turns after the wonder is built, the population on the tile is considered to be (15 / (15 + 10)) = 60% Newfaxian and 40% Oldham. The borders change and the tile is controlled by Newfaxian (The borders change when Newfax has the majority of population on that tile).
If the city centre tile flips (Original owner loses majority in the city), control of the city would go to the civ exerting the pressure on that city (So basically what loyalty does now).