There are several scenarios where a dispatcher can divert a flight once airborne. As PIC I am responsible for the safe conduct of the flght but I do so with the dispatcher acting as the extra crewmember of the flightdeck. He has a better picture of downline weather or other destination conditions that may not make it a prudent choice to continue on to the destination.
The PIC has the final say in most respects but it is a fool that doesn't make the dispatcher part of the team.