Kit?
I don't clean barrels. Will Shuemann is correct.
I do chambers, extraction and that is all I concern myself about, and mags.
Auto Parts store has these wire brushes, with a loop on the end, smush loop to through barrel "if" choose to shove through with a patch.
Twist and turn to clean chamber.
I most times use the one bent, like a shotgun chamber brush, and rack slide, twist and turn chamber, hit the extract and use a drop of Kleen-bore Formula 3 from out of a itty bitty Tabasco sauce bottle.
Pipe cleaner and that itty bitty bottle means I don't have to use that brush.
Fits in a pill bottle, handy, works, and this and glass pill bottles, or old shoe tins are what we did , mentors did in the mid 50's when I was born.
We got guns to shoot, not clean.
We'd run 500 rds a day or more , each day of the week, cleaning cut into shooting and reloading time.
Gal of Formula 3 we had to go to, as Browning quit making them tin cans of their oil we liked for clean and lube.
Shooting lead means Lewis Lead Remover or brass wool, copper wool and punching through chamber.
Get the lead load right, and not even worry about lead ...