I use cotton patches, usually cut from old socks or t-shirts, doubled over, just wet the edge or a small portion of it to start. Re-wet as needed when it begins to dry out or get used up. I usually dye one color in an area, then dye the next area in another color with a clean patch of cloth, then sneak that patch gradually into the other color, rubbing/blending them into one another, dipping conservatively into the appropriate color needed to get the effect I'm after. I can't really tell you exactly what to do because it's a cause and effect 'in the moment' type thing.
0000 steel wool comes in handy too. It can be used to fade and lighten colors which compliments the blending. Sometimes I'll dye the bow considerably darker than intended, then use steel wool to vigorously work certain areas, removing a majority of the dye from the surface and summer wood, but it stays in the pores, grain, and spring wood. This is a nice effect. Steel wooling less and less back toward the handle or tips can cause a nice 'blending' effect of its own.... 'blending/fading' dyed areas into almost natural wood color with just steel wool.