Over time, any petroleum based lubricant will erode wood. I do not know about the synthetic lubricants but I would suspect they will do the same thing. The cellulose fibers in the wood become soft when saturated with oil. It is best to store guns with the barrel down so the oil will not drain into the wood. A light coat of sealer finish is also a good thing over the often unfinished internal areas of a stock.
However, if you use a Precision Fit Stock as I do, you have little worry.
I have worked on many stocks where the wood has become soft and stained from the oil. ... Its good to put a thin sealer on any bare wood. If you really like to use more oil maybe you could or should store your gun barrel down so the oil runs away from the wood. Just a thought, PJH
Most guns get overlubricated and it WILL eventually soak into the wood if the wood is unsealed. I remember years ago when an old gent brought a beautiful old SXS shotgun into the gunshop I worked at that had a broken buttstock as a result of a lifetime of over oiling. It was a damascus Holland & Holland 12 bore that was otherwise exquisite.