Oil is fine on an airplane either checked or 100ml carry-on (checked is probably better in case the agent says no to carry-on), but the alcohol based lubes with the flammable warnings are not allowed.
https://www.tsa.gov/travel/security-...s-and-vinegars
https://www.tsa.gov/travel/security-...-aerosol-paint
I just got a bottle of Pedros Chainj, and it's fairly light and fluid, but not too runny, and still sticks well to the chain. The bottle has a very small nozzle, much finer than the Finish Line bottles, so it's easy to control small drops.
I've only tested it for a couple hours, so I'll report later how long it lasts in wet weather and how dirty it gets.