I have experimented with wood dyes that dissolve in alcohol.
I needed a blue green tone, so I ended up using a mix of blue and yellow
dyes. I applied to baltic birch plywood with a sponge brush. The results would have been fairly homogeneous even for birch which is known to be difficult to stain. However, the wood acted like chromatography media and there were areas that were green, areas that were more blue and others that were more yellow. Not really the effect I was going for but still looked good. I finished that project with crystal clear water based poly urethane.
I later tested the dyes on magnolia wood that I milled myself and that wood did not take up the dye as well. As to economical, the sponge brush can be squeezed out and cleaned with alcohol. When I apply a finish with a “rag” it is only about 4 sq.in. and I wear gloves.
I have seen pictures of some spectacular results from dying wood. However, I think that it must take a lot of experience and the right wood to get these picture perfect results. I also don’t like the way the alcohol solution raises the grain in the woods I tested.
That being said, I have been using Minwax water-based semi-transparent stain with excellent results. They have a pallet of 200 colors and can do custom colors. The stain is like a thick paint that you brush or rub on and then wipe off. The results are excellent in my opinion and much easier to use than dyes. You can still see the grain, but maybe not as well as with wood dyes. I have not seen any segregation of the color components, probably because the pigments interacts stronger with the stain components than the wood. The only drawback is that it dries rather fast for a water based material and it need to be wiped off quickly to get good results. I got good results on 12"x20" sheets of birch plywood, except for the edges where I was holding the piece. For laser cutting, the edges don’t matter much.
I Just finished a dye job on a rather large piece and I would recommend a sponge brush and rubber gloves. Anything the sponge absorbs can be smooshed out with your fingers by pressing the sponge into the wood.