Completely unrelated, in that they've never passed through the mixer? Then, what does the mixer do with the coins you've sent to it?
It's difficult to know what they do with the coins, but they could be selling them on an exchange to move onto a different currency and then buy bitcoin again.
Nobody's bothered doing this yet on any kind of scale, but that is only because major players in the underground Bitcoin space have found other ways to get caught.