Unless they are in an emergency and are busy with aviating, they will coordinate with their dispatcher on diverting, even if only to verify that the weather at the intended alternate is still favorable. Per the FAA regulations, the PIC and the dispatcher have joint operational control over the flight. Of course, at the end of the day, only the pilots have their hands on the controls, so they can make the plane do what they want—but from a legal standpoint, the dispatcher and pilot-in-command have equal & shared responsibility for the safe operation of the flight.