It depends on what kind of stain you use. I would not use oily/waxy stuff like Minwax, etc for fear of delamination. I've used aniline dyes with no issues whatsoever. They come in different versions that can be mixed in alcohol, oil, or water.
I prefer the alcohol-based anilines because they dry very fast and add no moisture to the wood. You can stain them and glue them up later the same day. I mix the powder with Denatured Alcohol which you can buy in most department stores and paint stores.
Oil-based can cause glue adhesion issues.
Water-based can warp and twist your lams... and of course, it adds alot of moisture to them, so you'd need to completely dry them out afterwards.
Regardless of which you use, after applying the dye, go over them with a stiff nylon brush, then a soft bristled brush, to remove any dye residue.
I only bother to dye the two lam surfaces that show under the clear glass, and then the edges of the bow after it is shaped and sanded.
You can find aniline dyes at woodworker supply companies such as...
Woodworker.com
Woodcraft.com