Well, actually, we at ASA used to, and in a way we still do.
We used to require 3000TT to upgrade, but that was dropped when they were really short on Captains a little while back. (Worked well for me!)
However, for the CRJ, we still do have upgrade mins. They are ICAO mins, stemming from the fact that we run the airplane international. I believe it requires either 1000 or 1500 PIC time to upgrade. SIC time counts at 50% (ie 1000 SIC = 500 PIC) I'm not 100% sure about the exact time required, since when I filled out the form I met the requirement and didn't really pay attention to it.
As previously posted though, most of this will probably be irrevelant as upgrades have pretty much come to a screaching halt. I believe CRJ Captain and ATR Captain are now going 3+ years, and E120 Captain isn't moving at all.