Thanks for sharing Mike Sherrill's methodology.
I don't know where this silliness about not pulling a brush back through the muzzle comes from... Top shooters ONLY pull the brush back, NEVER push!
Not attempting to be provocative and not referring to anyone posting on these pages, I would hesitate to suggest categorically that all top shooters only pull the brush, never pushing it. There is variation in methods used by any number of champions and it's not necessarily the measure of being a top shooter nor how they got to be that way.
The chief reason why some shooters, perhaps many,
pull the brush through the bore rather than pushing it, is that this is the best way to prevent the rod itself from touching the bore should it bend even slightly under pressure. To safeguard the crown, it would be advisable to fit a muzzle guide for the rod. Without such a cap the crown would be at risk for damage from the rod. A damaged crown is not good for accuracy.
A very knowledgeable and respected Canadian posting on Rimfire Central explained in 2016 what is involved:
There is also the option of a fitted muzzle guide cap for a rod. It does the same thing a bore guide does, but at the muzzle end. Anyone with a lathe can quickly make you one. I only pull rods from breach to muzzle. Its the only way to absolutely prevent contact with the rod and the bore. A few years back, Mr Stiller did an experiment on numerous top brands of steel cleaning rods using a support fixture and dial indicators and found that even without any pressure, they all bow under their own weight enough to contact the bore. (I don't believe he tested carbon fiber rods for this) If your pushing them, you will definately swipe the bore with some presssure. Not a big issue if the rod has no steps, is straight, hardened stainless steel and highly polished. Can eventually be an issue otherwise.
When I see many older rifles that have been regularly cleaned using rods, one common issue I often find is a a bit of deformation on the lower side of the crown. This happens from pushing out a jag, brush or patch from the rear and when the rods working tip clears the muzzle, the bare rod drops down on the lower side of the crown. No, a muzzle guide does not prevent this. This can easily be avoided by cleaning the rifle in a verticle position, but most people I know still do clean with the rifle horizontal.
Post #11 h t t p s://www.rimfirecentral.com/forums/showthread.php?t=735801
For those unable to find or fashion a muzzle guide, it might be best to use a cleaning rod from the breech end with a good fitting bore guide.
As for pulling a brush or jag back through the bore after pushing it through, rifle makers like Anschutz recommend cleaning in one direction only. The very practical reason for this recommendation is to reduce the risk of damaging the muzzle/crown. Gunmakers no doubt prefer to minimize the chances of damage caused by cleaning that is not done as carefully as possible and so recommend shooters to clean in one direction only, from breech to muzzle. It's simpler and more straightforward that way.
In addition to this cautionary advice from rifle makers, when going from breech to muzzle there's the satisfaction of knowing that the cleaning is pushing fouling out of the bore rather than pulling it back toward the action and trigger area.
No doubt many shooters have read on the internet that "more damage is done to rifles by cleaning than by shooting". Like many things on the internet, it may not be entirely true. Cleaning itself doesn't damage the bore. It's over-aggressive and over-zealous cleaning done without using the proper tools. If using a cleaning rod, use a bore guide. Never force anything. Don't go bat crap crazy on cleaning. Do it with care and due caution. In non-competitive shooting, a .22LR barrel should outlast it's first owner.
In the end shooters should clean their rifles as needed and using methods that give them positive results.