Maybe, every Karin is a part of the original, that she doesn't want anymore. Her wish would be, that these indesirable aspects of her disappear. Yuu's Tsukumogami then catches them and uses them, to fulfill Yuu's wish.
The original goal was, to have Yuu take one, and having thus his wish granted, while the other aspects would simply pass by, using this situation as their farewell.
But Yuu being that much in love, led to the Tsukumogami to change his plan, and rehabilitate the Karins.
PS: Aika obviously knows much more than she lets on. Remember how antagonistic she was towards the other Karins since the start.