I have done it several times. It works fine....especially if there is a dark line at the transition that will resemble landscape figure once it is dyed.
I use water soluble aniline. The three main colors for this application are walnut, brown mahogany, and red mahogany.
On this one, I used black dye, simply because I liked the way it looked with the dark striping in the wood. The sapwood is along the center joint, and tapers to about 3/4" wide at the butt.
If you use a light-colored backstrip like this one, be sure to seal it with shellac or lacquer before applying the stain.
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