I suppose it may differ from type to type, I fly Q400 and for us the computer schedules the cabin climb rate based on the actual rate of climb and differential pressure. Generally on a normal climbout we see cabin to climb at a rate of about 400-500 feet per minute through-out the climb. The computer initially keeps the cabin altitude low and allows the maximum differential pressure to be reached just above 10k feet, because or initial climb rate at low altitudes is high. Once we are higher up we can no-longer climb as fast, so the computer has a chance to catch up with the pressurization. It does not pressurize the cabin at 8000ft, it uses the differential of 5.46 psi instead, and based on this, your cabin altitude can be around 6000ft if you're flying at FL200 or 9000ft+ if you are above FL250.. (not certified, but just to illustrate a point )
It is a bit more tricky on descent, because q400 is not as slippery, like a jet, so you can, if needed, drop like a rock - in these cases you can notice the pressurization computer trying to guess what we are trying to do - and if it gets its calculation wrong, it will drop the pressure at about 1000 feet per minute or more when you're just 2000 or so feet away from the pre-selected landing altitude. It can get unpleasant for the ears