I have done many gun stocks over the years, well over a hundred. Lost count years ago. I certainly do not consider myself any sort of an expert on the subject, just well experienced. Walnut is an open grained wood and to be finished properly those open wood pores need to be filled. Why, to help keep dirt and moisture out of the wood pores and to make the surface of the wood smooth.
Only one fella has mentioned filling the grain, and that was with egg white. Well I have never tried egg white, might work like a charm, I don't know. But at least he is filling the grain of the wood.
I have two methods of filling the grain. One is to put two coats of a drying oil on the wood such as Lin-Speed oil (linseed oil with dryers) or polymerized tung ( tung oil heated with dryers added) after each two coats reduce the finish to the wood surface with #0000 steel wool. This will cut all the dried oil off except whats built up in the pores. Repeat this process as many times as it takes to completely fill the pores 100%.
The other method is to sand the wood stock with 220 grit or finer paper leaving the wood dust on the surface apply oil with two finger tips creating a slury of oil and wood dust. Rub in all around an into the wood well. Let dry for several hours, it will stiffen up, sand some more creating more slury, add more oil a bit at a time. Let dry. Repeat process a couple times. Let dry several days, until it all hardens up. Cut it down to the wood with #0000 steel wool. Pores should be full, if not use method one until it is.
Now the final finish can be down with an oil with dryers. Polymerized Tung, Lin-Speed, Tru oil. to name a few popular ones. OR you can final finish with pure oil, Tung, or Linseed oil. Tung being the more water resistant of the two oils. Pure tung oil dries flat with little sheen and can dry wrinkled if applied to thick. Linseed is easier to work with, more forgiving but is weaker less water resistant finish. Note: there are as many receipes for gun stock oil finishes as there are chocolate chip cookies. Thick books have been written on the subject. I have many of them and it does get very confusing.
I am an oil finish person. A decent finish takes time, and patience to apply. Many finish coats need to be applied by putting a drop or two of your chosen oil on your bare finger and rub it around. You do not have to rub until you create heat, just smooth it around evenly, then let it dry.
Pure and Polymerized Tung can be had from Lee Valley tools. Birchwood Caseys Tru Oil where ever. Lin-Speed Oil at this time is not available in Canada. If someone wishes to try it PM me and I can get it for you.
My two favorite products that I keep on the shelf for doing stocks is pure and polymerized Tung Oil and Lin-Speed oil. There are hundreds of other combinations of oils and dryers and methods of application. But this is what I have used over the years and in the past with success for me and the folks that I have done stocks for.