I stay away from Rem Oil for surface protection and actually use it as a cleaner for the bore if the shooting session was light and also to clean primer residue from the firing pin housing area.
If you notice the warning on Rem Oil, its very flamable. While most oils will burn, if there is a warning on the side of the bottle for flamability that usually indicates a pretty serious amount of strong solvent content. This solvent may serve as a catalyst or carrier for the teflon constituents in Rem Oil. This solvemt may be harmful to wood, plastic, rubber or a gun's finish over a long period of time.
I suggest you go with CLP,Beretta Oilio, or Benelli Oil. They are gentle yet effective for surface protection and lubrication. Many times they can be utilized as cleaners as well.
I cannot speak for Bore Brite. If you use a Bore Bright or Hoppe's Number 9, make sure the solution drys or flashes off before applying your final light coating of lubrication so that the two do not mix. As has been previously mentioned on TFL, a good thing to do is to clean the bore and then put the barrel aside for a few minutes and clean the frame of the pistol or do something else while the solvent drys or flashes off.
Good Luck
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"When guns are outlawed;I will be an outlaw."