Maybe it's just the USMC training, but I rinse the barrel and dry it at the car, and dry off the lock too, before taking the game home. So is that "cleaning"? When I get home I hang up the game, and then scrub the rifle or shotgun bore, dry it, and oil it. The gun or rifle's exterior gets a rub down with an oiled rag, then the gun is put aside and the game is tended.
The small game isn't going to spoil in the 30 minutes I take to scrub, dry, and oil the gun, as it's cold out, and if I'm butchering a deer, it gets dressed where I hunt, the body cavity is propped open, rinsed, and the carcass is loaded onto the vehicle. The gun is then rinsed and dried...the drive home then begins..., so is that game before gun? At home the deer is hung up to cool, and to allow rigor to set in and then subside....or it's dropped off at a the butcher to process. Depends on what I want to do.
I've hung deer overnight in temps under 40 degrees and butchered the next morning. Now if it was above 50 degrees, or it took me a while to find the deer (I've had to track a few for friends who use modern guns..., I point out the no need for tracking when folks use a sidelock and a round ball...but they don't listen)..., then I'd do the deer immediately, and get it cool and clean.
I also check the gun the following day, to be sure my rust prevention has succeeded.
LD