For short term storage I use a light coat of Rem Oil in the bores & a wipe down with Rig on the exterior. For long term storage I use Rig in the bores as well as the exterior. I've never found Rig to run on any occasion
The reason for Rem Oil in guns that I'm going to be using is that if I forget to put a patch through the bbl before shooting it's not a problem.
By light coat of Rem Oil I mean a little sprayed on a patch & pushed through the bore once & pulled back out at the breech end. I have found that even with this light application assembled guns stored muzzle up will have a trace of Rem Oil on the breech face within a few days from drainage down the bbls & that can't be good in the long run as it will eventually migrate down to the wood.
I don't know if this is true with all light oils but it's muzzle down storage for me or guns broken down & stored in their case.