Your links aren't working. Without seeing what your working on its hard to make a determination. I always try to avoid petrochemicals on wood as they have an extreme drying effect and can cause splintering.
I use a heat gun directly on the trouble spots. Do not heat it enough to boil the oil still in the wood. The oil will rise and bring the grime with it. I then wipe clean with cotton flannel or something equally as soft then repeat until as much comes off as I can get. As a side benefit the heat helps raise some of the lighter dings in the wood that fill with oil and grime. Avoid getting heat near the cartouches because it has the same effect on them and they will raise and disappear.
I avoid using sandpaper if at all possible and prefer to work with steel wool. Generally 3/0 and 4/0 and very light pressure. You are much less likely to cause gouging with steel wool than sandpaper and once you gouge you have to sand more to remove it and the viscous circle begins.
If you don't have a heat gun a hair dryer can be used it just takes a little longer.
Also if no heat gun or hairdryer is available steam can be used as well but again without overheating and avoiding the cartouches by safe distance.
I also use steam to raise deeper remaining dents once clean again avoiding any cartouches or markings which there are no shortage of on
British rifles.
Good luck, fix your links and keep us posted on your progress.