I use Parker Hale jags with wrapped patches, and also nylon brushes, and I run them in both directions. With the jag/patch, I stop the stroke when the tip of the jag has exited the muzzle, but before the entire wrapped patch has exited (ie. half-way out the muzzle). If you let the whole thing exit, the patch may try to ball-up at the muzzle on the return stroke and be excessively hard to return.
With a brush, you pretty much have to let it completely exit, because if you don't, the bristles have to reverse direction within the bore and that causes difficulty. I let the brush exit, then slowly let it re-center itself in the muzzle on the return stroke before bringing it back to the chamber end. I can't see how you could possibly do any damage to the crown if you are gentle (as above) when you are transiting through it in either direction. I DO, however, always finish the final stroke of either patch or brush on the way OUT of the barrel to push any dislodged debris out with it, and I finish up with dry patches in the same one-way direction to push everything out the muzzle. I have not experienced any premature loss of barrel life using these techniques, and I believe that going in both directions does a better job of scrubbing the bore clean.
