This is a very type specific question. I fly the 737. Cabin altitude ROC is controlled by the cabin pressure controller. On the Classic at the maximum ceiling of FL370 you have a cabin altitude of 8000 feet and on the NG you have a cabin altitude of 8000 feet at FL410. The NG structure is capable of sustaining a greater differential pressure.
Depending on the system installed, at the beginning of the takeoff roll the cabin is already slightly pressurized or begins to pressurized past 60% N1. On the climb out, the controller maintains a proportional pressure differential. Sometimes if you are light you climb too quickly that you reach the maximum cabin differential pressure (8.65 psi on the Classic and 9.1 psi on the NG) and the pressure relief valves will open.
When you reach cruising altitude, the controller will modulate the outflow valve to maintain the lowest possible cabin altitude depending on differential pressure limits depending on the cruising altitude.
Upon descent, the controller will do the opposite, control the cabin altitude ROD proportionally.
You can find some more information
here.