Don't know if I'm looking in the wrong place, but finding out what they consider to be franchised and non-franchised cross-border services isn't easy.
I've found
this file which lists a number of services, including things like the 218 and X17, as being franchised but operated under a service permit. No date so I can't tell how recent it is and whether things may have changed since.
This would imply that franchising may have an impact outside the SYMCA area - for example, if the 43/44/50/X17 are franchised services and Stagecoach don't win the franchise then that will complicate operations in Chesterfield considerably. (Including the fact they were awarded ZEBRA2 money for vehicles to operate these services!)
I must admit I always thought these would be non-franchised services operating under a permit.
One positive thing I found in my search was a comment that SYMCA are committed to working with neighbouring authorities on service provision and ticketing, so maybe it's not as bleak on that front.