I've done two builds using dye. One was water-based, the other alcohol-based.

The water-based one (ash) I finished with wipe-on poly, the alcohol-based one (maple veneer over basswood) I finished with tru oil.
In both instances there was transference of dye to the cloth with the first couple of coats. The ash body would have dried for at least a week IIRC; and the maple veneer/basswood (with binding) would have dried for at least several weeks due to another build going on.

I should also mention that I went over both with a very fine synthetic abrasive pad and wiped them down with a clean dry cloth before applying any top coat anticipating that there may be some lifting of the dye (and also to knock back any nibs of grain lift from the moisture of the dye) and it still happened.

That being said, I think it's just part of using dyes. In both my cases they were single colour so not a burst or transition of colours, so it was not an issue. I didn't have any problem with uneven colouring either.

If it were me in your case, I would wipe my tru oil in straight lines going in the direction of lighter colour (the pink) to darker (the purple) for the first four coats or so (or until you stop seeing dye on your cloth). Don't go in a circular motion or you risk just smudging the two colours and not getting a nice transition/fade effect.

Just my 2 cents.