I have a Hoppes rifle/shotgun cleaning kit that a patch holder small enough to push through a .22, so I know they are out there.
Another option is something called a bore snake, which is a soft, braided, rope like thing with a built in bronze brush. The "rope" has the same surface area as dozens of patches, so you get lots of result for minimum effort. I run mine through the barrel once or twice (with no solvent) after shooting and end up with a spotless bore.
Objections that I have heard about the bore snake usually have to do with re-using the same material over and over again. But I must point out that it is washable, either by hand or in the washing machine after placing in a net bag. Others have posted that even if you did end up pulling some kind of debris through the bore along with the snake, the snake wouldn't push hard enough against the bore for the debris to scratch it.
One other option for pulling patches through that I have seen is to use a length of the plastic cutting strap from a weed eater. You can cut a length a little longer than your barrel and make a little loop in one end and then use a flame to melt the loop closed. Sorry, I can't think of a better way to describe it. You use the melted plastic as the adhesive to make the it stay in a loop. (I better stop trying to explain, or i will end up confused!) Then you can just stick a patch through the loop and pull it through the bore.