I don't know if this will be of any help or not. I had a custom millwork and cabinet shop and would sometimes run in to this with white ash. On case goods, I would finish first - then address the holes with the correct color nail head filler putty.
The thing you have to remember is it depends on the final finish you are going to put on the stock. I also used to teach shop - back when everyone was teaching mixing glue with sawdust, etc. The problem is, the glue will not absorb stain and it will stick out like a sore thumb. You don't state if you are going to stain your stock or not . . and if you do . . what type of stain and what type of sealer . . i.e. BLO, varnish, etc.
Another thing I'll mention is that to me, and this is my own personal opinion, each piece of wood is a "natural wonder" - nothing is perfect. If I was using ash, I guess it would depend on how bad the tunnels are, where they are located, etc. Personally, if they weren't overlay "offensive", I'd probably be inclined to let them be as it's part of what you get when working with that species and it just adds character to the finished piece.
Ash, like a lot of other species, can vary in density and each piece will accept stains differently. If you fill with a "dark" filler with the idea that the stain you are using is supposed to be "dark" .. . . you might be surprised to find that your wood, when stained, takes on a lighter shade than you intended - thus your repairs "stand out".
I just mention these things as they are things to consider. I hope you'll post photos of your rifle when you finish it - I'd love to see it! I like southern rifles and I've seen some nice ones stocked in Ash. Ash is a nice species to work with. Whatever you do, it will turn out just fine. Good luck and I hope you'll post some photos!