I carried a Chief's Special for 40 years and I treated it a little different than most Revolvers I own because it had to be utterly reliable and never have oil coming out of it which could compromise ammo or my clothing. I will describe what I did.
After shooting it, I always removed the Cylinder to clean it ( a single screw). After the Chambers were clean I would use Rig #2 Oil and fill up the Ejector Rod Tunnel, carefully operate the unguided Ejector Rod a few times and dump the oil out onto a rag. I would usually repeat this 2 or 3 times and then the oil would come out clean. I then blew any excess oil out with an air compressor to leave the tunnel relatively dry, but a slight bit of lubrication and rust preventative would remain behind. There was never enough left behind to compromise ammo or clothing after the cleaning process. Rig #2 Oil was a fantastic product but sadly when Birchwood-Casey was sold the new owners of the Company decided to discontinue it. The next best product I can think of (for this purpose only) is Remoil - because it is thin and evaporates very quickly. Yes.... Remoil! I am not recommending it for any purpose other than this and if blown out with compressed air will never gum up, but yet will leave little protection and lubrication behind in a dry state. I would not recommend using a heavier oil like CLP or any that does not evaporate quickly. I also like the ability of the Remoil to loosen up unburned powder and debris from inside the ER tunnel. Being thin also allows for easy removal and quick drying with compressed air.
This procedure allowed my Chief's Special remained clean, have a free spinning Cylinder and always operate properly without having oozing oil from the Revolver's Cylinder. Ammo never had any oil on it and gumming up or retarding of the Cylinder's rotation was never an issue. Every few years after extensive shooting I would disassemble the ER from the Cylinder to fully clean it. Quite honestly it had never gotten that dirty due to the regular cleanings of the ER Tunnel.
Sadly, the Rig #2 Oil is no longer available, but the Remoil works just as well for this purpose and is available pretty much everywhere. Just make sure you bow it out with an air compressor that does not have moisture in it or canned air.
I'd not do this with heavy oils!