Do airliners carry radios that switch automatically (as has been intimated by some pilots I have spoken to) or do these radios require pilot input? If so, what triggers the change?
Automatic transfer of comms, not a good idea!! The transfer oc comms, as explained by DFC, is carried out by the controller, based on several factors, such as clearance from other traffic etc. There is no way a magic box on board an aircraft could know the thoughts and plan of the controller, maybe he wants to keep the aircraft beyond the Transfer of Control Point, after agreement with the next unit. For the pilot to be automatically sent to the next unit before the agreed time would give an assumption of handover of control, and frequency separation is not a safe way of doing things.
Also, frequencies are regularly out of service for a variety of reasons, or can be subject to the collapse status of the sectors within the unit. All this would be unknown by anyone except the controllers concerned.
We do have Datalink with some carriers, whereby we can send silent messages to aircraft via transponder giving the next frequency to the pilot, but again the frequency (or channel) is specified in the message.