I do not use brushes on my 22s anymore. I use a brass jag and patches. I tried a couple different jags and realized that certain jags work perfectly with certain patches. It's a trial and error process but you'll figure out which jag and which (thickness) patches.
Now cleaning 22 barrels is a snap for me. I think that cleaning brushes are a generic substitute to use on barrels (even other calibers) until one finds a good combination of jag/patch which fits the bore of the particular arm, then brushes are no longer needed at all.
I clean my 44 Mags the same way, no brushing. A proper fitting jag/brush will get lead out of the barrel too. You'll know you have arrived at the correct combo when your patches come out displaying rifling marks. Copper fouling is better removed with a good jag/patch combo and a copper solvent, rather than a brush in my experience.