Transfer of control takes place at a pre-determined point agreed between the ATC centres (ATCC). Info on the flight is passed normally by electronic means at a defined time prior to the aircraft reaching the transfer of control point.
The agreed transfer point will not always be on the FIR/UIR boundary and can be some distance from it.
The required separation and when radar separation can be applied is also agreed by the ATCCs in advance and written into their procedures............eg as far as I remember, traffic at the same level exiting many ATCCs use 3 minutes separation but this can be reduced to 10nm Radar separation in most cases without the controllers having to talk to each other.
If the electronics are broken then the controller or their assistant has to pass the info by telephone....hence slower traffic rates when this happens.
Transfer of communications is a related but separate issue and happens mostly before the flight reaches the transfer of control point..........i.e. you can be talking to the next ATCC but still under the control of the previous which means that if you request a level change the people you are talking to have to either wait until you pass the transfer of control point or ring the other ATC and ask.
Some of the modern FMS/ Radio controllers use the current aircraft position along with the Jeppesen database to give the appropriate frequency to call on...........but the pilot has to make an input to change the frequency.......and if told to use another can dial it in.........................even a cheap handheld GPS with an aviation database will often provide the frequency for the airspace.
Regards,
DFC