When I adopted two adult cats last year, I initially secluded each in a different room upstairs (at the advice of the shelter). This left me with your problem: after the cats had access to more of the house, I wanted those boxes to be in the basement. (Mine aren't behind a cat door, though.)
I asked the folks at the shelter (Animal Friends in Pittsburgh PA) for advice on making a smooth, accident-free transition, and they said a drastic change would confuse the cats and I shouldn't do that. Their advice was to move the boxes about a foot a day. When I got to the steps I obviously couldn't maintain that (couldn't prop it up on a step), but what I did there was to move from the top of the steps to the landing in one day, to the other side of the landing the next, and to the foot of the steps the next day. From there I continued the foot-or-so-a-day progress toward the final placement. I did pick up the pace toward the end, more like two to three feet a day, and they seemed to do fine with that. This leads me to suspect that the advice I got was overly conservative, so you could try faster movement. I did make sure that the box's placement after a move was always within sight of where it had previously been.
I also had another box in the basement the whole time, but they didn't start using it until near the end of the move.
Finally, I would recommend moving the litter boxes first and only then introduce the cat door. Can you leave that door open during the move? (I say this because you don't want to place any barriers between them and the boxes, and a door they don't know how to use is a barrier.)