I have shot .45 Colt black powder loads in cowboy action matches. Most of the time I used a repro 1873 Winchester, but I shot a few matches with a Model 92.
You will get more fouling blowback with a .45 Colt compared to the thinner walled .44-40, but the .45's fouling is manageable. Use heavy bullets, not under 200 grains, large powder charges, and crimp the bullet into the case as hard as you can. These recommendations will tend to increase chamber pressure, blowing the case out to minimize blowback.
Use a clean burning powder or substitute. In my opinion, Goex is one of the dirtier powders. I would use Schuetzen for real black powder, Pyrodex for a substitute. I am not interested in hearing about how corrosive Pyrodex is...I find it burns cleaner in my cartridge loads and I am willing to do the cleanup required by its use.
I recommend prepping the rifle by greasing the internals liberally before shooting. It's like spraying your cookware with PAM...makes cleanup a lot easier after shooting. If you are shooting a lot in one session, keep the fouling in the action moist with aerosol Ballistol, or Ballistol mixed 1 to 10 with water, or PAM cooking spray (it's canola oil). Make sure you have a compatible lube on your bullets; SPG, crisco/beeswax, etc. Some blackpowder substitutes allow use of smokeless lubes on your bullets. Investigate. I use home cast bullets dipped in crisco/beeswax in competition.
Black powder fouling in a repeating rifle cleans up just like black powder fouling in a muzzle loader. Hot soapy water can do the job...just make sure you get all the nooks and crannies dry. I use lots of Q-tips when cleaning a rifle. I have never disassembled a 92. '73s are much easier, and tend to be more reliable in a cowboy match. But when I used Pyrodex .45 Colt cartridges in my '92 I was able to flush and clean the action without disassembly sufficiently that I never had rust or corrosion.
Probably the easiest way to clean the action is to take the stocks off and spray the internals with aerosol brake cleaner. (Brake cleaner can damage a stock's finish.) It does a great job, fast, dries quickly. Be sure to wear gloves and eye protection because break cleaner is nasty stuff if it gets on you.
Be prepared to remove the magazine plug and clean out the magazine and spring and follower maybe once a year (if you are shooting a lot). Greasing the inside of the magazine tube is a good idea too.