I don't tend to clean .22LRs as much as I used to, even my benchrest barrel, which is a Lilja (stainless). Stainless steel is softer than chrome moly so many competitors change barrels after about 15,000 rounds.
With competition barrels, it's good to clean them more often when new because they sometimes lead up a bit in the Leade area until fully broken in. Some shooters use old brushes (cleaned with brake cleaner) to scrub the lead out of the lead without pushing the brush further down the bore.
Stainless barrels break in sooner than chrome-moly, but quality chrome-moly barrels can outlast several stainless barrels used for rimfire benchrest.
Regardless whether a barrel is chrome-moly or stainless, wax on LR rounds will protect the bore, often better than cleaning solvents. Whenever solvents are used, they should be followed by a patch with preservatives like Break-Free. Most solvents don't contain preservatives.
In any case, clean from the breach end whenever possible, use a rod guide, and try to get a cleaning rod that is no longer than necessary to push the patch out of the bore. The rod should be stainless steel and polished to a bright finish, so it won't pick up grit. It should also be wiped off repeatedly so any grime on it won't wear the rifling, especially at the crown.
I realize that most plinker shooters won't want to go through all this trouble. My rifle bores get treatment commensurate with their accuracy levels.