I've stained some. I did one maple bow with Minwax stain. It felt very waxy afterwards, but as far as I know it's still shooting... although I don't think that guy shot it much. I wouldn't use that stuff again.
I used water based anilines, and it worked ok. I had to wait until it was dry and the lams warped a little, but it didn't seem to be an issue.
On the rest I've used alcohol based aniline dyes without a single issue. Don't worry about saturation. It's alcohol, it evaporates quickly. I use a cotton cloth cut from old t-shirts or socks, get it wet enough to get even coverage. Give the alcohol plenty of time to evaporate, then go over it longways with a clean very stiff bench brush to remove any cloth particles or dye residue.
The last one was curly maple that I put in a one piece recurve, and I mixed up a 'weak' dye, just to tint it the tiniest bit of a light rose color, but more to make the curl really jump out at ya... and paired it with a cocobolo riser. I hated to see that one go.