Hoppe's #9 is a bore solvent. It is made specifically to dissolve residue from the inside of the barrel. It is NOT a lubricant. When I use it, I make sure it is removed completely before I add anything else.
Break free is the military's CLP, for clean, lube, and protect. It does an .....ok job at all three. What we use it for in the army for cleaning specifically, if you have a dirty rifle, soak the guts in the stuff and let it sit overnight, it does a good job of drawing out carbon from the metal. But, if you say it's clean to turn it in, and then add CLP again, it will KEEP pulling dirt from everywhere in the rifle, you will never get it to stay clean. It's best as a lubricant, I used it for a long time after I got out of the army.
Now I use gunscrubber to remove carbon from non-plastic guns, when I feel that the bore needs cleaning I use Hoppe's, and I use Mobil 1 5w30 full synthetic for lube.