Only difference to the two is it tends to be darkened to make it look Walnut but otherwise it behaves the same.
You'll Shoot your eye out Kid
I have done 3 beech stocks, the first two went well but I wasn't happy with the last one. It had a lot of residual staining after removing the original finish. I had it sanded up nicely but tried the trick of soaking in bleach water to remove the stain. This kind of worked but the wood never took on the stain I applied successfully, maybe because of the bleach. I originally used Danish oil over the stain but refinished using a coloured wax. Still not what I was after but its OK I guess.
Plinkerer and Tinkerer
The Bleach was probably still in the wood effectively, it's a bugger for seeming dry when it's not quite, probably best to let it sit for quite a while after bleaching in a warm dry place so it goes completely dry.
You'll Shoot your eye out Kid
I have used oxalic acid to bring out the grain in a beech stock before oiling.
I recently used Parker Hale walnut oil on a walnut stock and lots of light coats and buffing in between has given a really nice satin finish.
Lots of sanding with very fine grade paper(2000-3000)is worthwhile before applying the finish.
Alkanet oil on beech:
My airguns: https://barx.org/airguns/