I've done several dyed objects using various dyes, Craft Supplies sells pre-mixed and powdered aniline dyes (water & alcohol soluble.) I have also used water soluble fabric dye - they take a 'long' time to dry - so I'm sticking to the alcohol soluble ones from now on.
After turning a couple of art pieces (Jimmy Clewes course & video) he recommended using a "Sharp" black pen around the boarder. With lacquer, shellac and waterlox, I've had no issues. With the Liberon Finishing Oil, just the black boarder where I used the sharp won't dry correctly - after two weeks, it's still tacky. The coloured areas dried nicely and didn't run. I'm going to have to remove the finish on the edges and I plan on using the black dye applied with either an air or paint brush for the edge.
Sad to say, I ignored the "test on a scrap piece first" and have extra work now.