I clean mine the same way that I clean my black powder rifles: water-soaked patches. I run a few through the barrel, then a couple of dry patches. I'll wet a rag and scrub a bit around inside the action, then dry it off and oil like normal.
No rust, no problems.
Well we don't rent pigs and I figure it's better to say it right out front because a man that does like to rent pigs is... he's hard to stop - Gus McCrae