There’s a thread from Michael Petrov about his process for cleaning rifles that’s worked well for me in the past. Similar to Steve’s post above, it calls for raw linseed oil.

From MP’s Cleaning the classics post:

“I'm kind of jumping ahead but every stock I have had to clean in the last decade I have used nothing but raw linseed oil, a tooth brush for the checkering and in some cases 0000 steel wool.

Some have been very dirty and a good coat, lather it on until the stock is wet allover. This will do wonders for cleaning the wood and not removing the underlying stock finish. Let it soak and take your time, wipe with a clean cloth or paper towel.”

Link to post:

https://www.doublegunshop.com/forums/ubbthreads.php?ubb=showflat&Number=351571&page=3

-Tom