It's right before it comes to a complete stop, there is a slight (but noticeable) shift, like it doing a little skip before right before stopping
Does it matter if the car is warm or not? The reason I'm asking is when the engine is cold, the idle is set higher. On the final downshift into first gear, that could cause a bit of a jump that would be more noticeable because of the extra engine speed. And because there's more engine speed (power), there is more work for the brakes, which might feel like they are slipping.
Another thing to consider, is it only in Sport Mode that this happens? Or any mode?
it does this both when I brake hard and brake gradually. But it does happen consistently.
Here's an experiment you can try: On a safe road, put the car in neutral while going your normal speed, then brake to a stop. This disconnects the engine and the transmission, so there will be no engine braking as the tranny downshifts. If you don't notice either the skipping or the braking issue, then I expect that you have isolated the problem-- The transmission is either shifting too quickly (not equalizing speeds correctly), or the engine is running faster than it should during downshifts.
If the brakes still fade a bit, then there's something screwy going on, perhaps with the Stability Control or Differential, I would guess. Try doing the experiment again, this time with the DSC turned off and see if the problem recurs.
If the same problems happen even in neutral, with DSC off then I'm afraid you've exhausted this particular Random Internet Yahoo's troubleshooting skills. But it's something you can tell the MINI guy.
Please let us know what you find out.