I am not sure, but I'll guess that it works by steps. This is because in in Colosseum you can put Shadow Pokemon in there who will then purify in the same manner to how they purify if you have them in your party - which is by steps. Leaving the game on does nothing to purify Pokemon, and seeing as that applys to those in the Day Care, I'm pretty sure that means the whole thing works on steps taken then. Plus seeing as everything else works as it does from the handhelds, there's no reason for them to have changed only one feature of it.
Of course, it's only an educated guess, but my money is certainly on it working on steps taken.