They will relocate once the new arena is completed/when the lockout ends, whichever happens earlier. Source: http://www.latimes.com/sports/sns-tsn-acn-name-njn-p1-20110926,0,2229297.story
The Clippers are the Clippers because of Donald Sterling, not because they are LA's other team. Not that the Nets have a great reputation for management, either.
If they made this move when it was supposed to happen, when the Knicks sucked, then it be different. But now the Knicks are the big team. Although, it wouldn't surprise me to see Brooklynites follow the Nets, especially if the Nets can build around Deron, and it doesn't hurt that their owner is Russian. Hey, look at the Mets. They may not be #1 in NY, but they still have a strong fanbase in their area.