In the case of that stock, I would tend to want to clean it without removing the finish.

Here's something to try. Get some el-cheapo no-name vaseline. Liberally smear it all over the exterior of the stock and let it site for a day or two. Then use lots of paper towel and wipe all the dirty vaseline off. Much of the dirt and excell oil will have leeched into the vaseline and wipe right off. Then hand-rub small amounts of BLOicon into the wood and buff off the excess after about a minute or two. More crud will come off on the buffing rag.

Repeat as necessary.