JB bore compound (I've read that there are 2 but I mean JB bore paste) was designed/developed specificly for cleaning rifle bores of copper fouling, by a champion benchrest shooter. It is a very mild abrasive. I use it sparingly on badly fouled bores like when I'm in the feild doing a lot of shooting, and need to clean quickly to regain accuracy. A couple of passes with JB following the directions is all that's needed. I don't use it all the time. After I get home I'll clean with a good copper solvent like Butch's or shooter's choice. It may take several days of nightly cleanings before I stop getting color on my patches, but too much scrubbing with any abrasive will shorten barrel life. After all if it's good for removing machining burrs, it's probably good for removing the nice sharp edges you have on new rifling, too.
I don't know anything about Flitz, but from what I've read here I think it's a little more aggressive than JB and was not specially made for gun barrels.