BP,
Here's a how-to I did a while back. You're farther along in the process, and really only want the Top Coat section. The type of finishing oil is not really important, use what makes you happy.
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Finish Stripping
I use the goopy heavy duty Methylene Chloride based stripper from the Depot. You have to have chemical resistant gloves, the stripper will chemically burn your skin. Let the stripper do the removal, not sharp tools. If your finish is the glossy epoxy based Browning standard, it will take 4-5 stripping go-rounds. I use cold water and scrub brush to remove the stripper, and a toothbrush on the checkering. All of the epoxy finish has to be removed, or it will show up later as wierd shiny spots in your oil finish. This is the shittiest part of the job and everything after this is fun.

Finishing and Sanding
First coat is a soaking sealing coat before sanding. I use Tru-Oil diluted 50% with Naptha. Feel free to use any good standard mixture instead of Tru-Oil, ProCustom, or Permalyn, etc. Coat the insided inletted area buttstock end too.

I'd put the buttstock back onto the frame, and replace the pad/buttplate at this point.

All of the sanding will be wet-sanding. For these pore filling steps I use pure Tru-Oil. I'd start with 220 grit Wet/Dry sand paper (the black type). Always use a hard backing to the paper or we'll end up with waves in the stock based on the differing hardness of the stock. Any stock with figure is more vulnerable to these waves than straight grain.

To wet sand, put a thin coat on the stock and wet the paper. Work on about 6 sq in at a time. You'll see a mud build up during the sanding. Wipe this mud across the pores during these early steps. Wipe off the mud completelely though, as it will dry to be a bit harder than the wood. Protect the checkering with blue masking tape, and keep the finish out of the checkering. Use acetone and a toothbrush to remove any finish that dries in the checkering ASAP.

Do the whole stock with a wet sanded 220. Let each coat dry completely, usually 2 days.

Decision Time - is it time to move up a grade in grit to 320? I don't know. The only way to tell is to move to 320, do a wet sanding, let it dry and see if there's any deep scratches that show up as coarser than the 320 scratches.

Going back is part of the game, and for me, I guess that half the time I move up a grade, I end up going back for more of the previous grade. Here's my estimate for go-rounds of wet sanding by grade:

Grit -Go rounds -Mixture
220 -1 -Pure T-O
320 -3 -Pure T-O
400 -2 -75% T-O and 25% Naptha
600 -2 -60% T-O and 40% Naptha

Top Coats
The stock is ready for top coats when there is no, none, absolutely no pits showing at the pores and no apparent sanding scratches visible. For topcoats I add a bit of boiled linseed oil to our Mixture - about 10% Linseed, 50% T-O and 40% Naptha. The linseed gives you a little bit more time before the finish dries. By this time in your project, you'll be mixing your own amounts
I put on a top coat with rag made from an old fine cotton dress shirt. The rag is folded into a 1" square with all the ragged edges folded in, not exposed. I dampen the rag with finish and use it like a finger held paintbrush. I want a very very thin and uniform coat. Try not to fuss with the topcoat (it's hard not to fuss). Let this dry for 2-3 days. When the topcoat is rock hard, do a very light polish with a Pumice/Linseed Oil mixture, using a hard felt pad dipped in the pumice slurry. Clean it off with clean rag dampened with linseed, then dry with a paper towel.

Another topcoat and pumice rub.

Decision Time Again.
Can we move from a pumice rub to a rottenstone rub? Same criteria as above. Rottenstone is much finer than pumice.

At some point you'll feel that the finish is done. Carefully toothbrush a 50% T-O and 50% Naptha mixture into the checkering.