It all depends on the situation. For aircraft inside controlled airspace and staying within CAS e.g airways, is mandatory to read back a change of frequency but I'm not sure about the station ID. If on first contact with ATC the aircraft doesn't use the station ID the controller will use their station ident.
For aircraft outside controlled airspace they may have a different route and decide to change to another frequency than suggested by the controller. In that situation, on leaving my frequency the pilot will more often than not inform me of the next station and frequency they wish to change to.
If there is traffic to affect OCAS the controller will inform them that there is traffic in the circuit at ???? etc and suggest you free call that aerodrome.
Hope that has helped.