Like others have said, a Lead free cloth will really help with getting the crud off of cylinder face and forcing cone on SS Revolvers.
Also, to protect your Muzzle Crown and accuracy, I wouldn't clean from the muzzle with a Rod & I would puchase a quality rod like a Dewey. Then, unscrew the brush to insert the rod & then re-attach jag & pull it through like a bullet from the forcing cone out & then re insert the rod & repeat. Sometimes you have to shorten the brush if you're using one for it to screw on. If you have a leaded barrel, use some 100% copper Chore Boy Dish Scrubbers and put a small amount around a brass .243/6mm bore brush and then pull it through your barrel. Letting the barrel sit overnight with some Kroil in it will help to let the Kroil penetrate under the lead and it will sluff off the rifling. THIS IS FOR Center Fire GUNS ONLY!!! RF Guns have delicate rifling and I would't do this with a RF.
With .22, I NEVER use a Cleaning Rod IN THE BORE only in the cylinder IF necessary. Just cut a piece of normal diameter Weed Wacker Line with wire cutter at an angle so its sharp enough to spear a .22 patch on & take a flame and melt a ball on the other end of your line to hold the patch on and clean your barrel by pulling that through your barrel and cylinders. On a SS Gun you can chuck up a brass brush on a drill and clean your cylinders out quick as well.
Again, on the Airweights I wouldn't use a power tool or a Brass Brush, only Nylon Bore Brush in Cylinders & work carefully by hand. That and a Stiff Nylon Gun Toothbrush for the Forcing Cone & underneath the extractor with some M-Pro 7 or the CLP of your choice.
I know you only have the .357 now, but you will buy more guns after that, I bet.
Good Luck!