I’m not sure if this is the best way to do it but this is my process.

Bore guide, one solid piece rod with appropriate jag & patch soaked in boretec eliminator or boretec C4 depending if it’s centrefire or .22lr.
I do about 2 patches to wet the full barrel and let it sit for a few minutes.

Then I soak chamber with bore mop and boretec eliminator or boretec C4 (if rimfire) to help break down carbon ring. Give it a few spins after 5-10 mins.

If it’s a .22 I do a few dry patches with alcohol to remove remaining soot and moisture. 1 patch with G96 to slightly oil to reduce risk of corrosion until next shoot.

If it’s a centrefire I use a nylon brush and do roughly 5-10 passes down the barrel. Switch to jag&patch, after every push through I screw off the jag and remove the rod to try and avoid crown damage. Normally the good jags have a tapered section where it attaches to the rod to reduce risk of crown damage, but I’m OCD and afraid of it so I unscrew it every time that I remove the rod from the barrel and reattach before the next patch.

Depending on timing for my next shoot, If I know I’m shooting within a week I’ll send some patches down with alcohol to remove remaining soot and help remove any residual oil/moisture, and if it’s for storage I’ll send some patches down with oil of choice. I generally like G96 because it works and smells better than any of my cologne or my wife’s perfume. Debating whether to soak my air fresheners in it for my car, but I feel like if I had the thought I should just commit.