My angle on this issue is as follows:
Training flight......instructor is paid for the flight so holds a CPL...but no AOC required because as a training flight it is not public transport.
If in moving the aircraft and student from the unlicensed airfield to a licensed airfield, the Instructor is paid then in my book, that initial flight is public transport since the pilot is being paid to carry a passenger to another airfield. Same goes if the instructor is a PPL instructing for free but the student pays.
Public transport flights require a licensed airfield.
Thus as far as I can see, the only way in which both instructor and student can travel in the aircraft from then unlicensed airfield to a licensed one is if they both do it for nothing...i.e. the student doesent pay to be a passenger and also the instructor is not paid for any part of that flight.
There is no law that says a passenger can not manipulate the controls. However they can't log the flight time.
If the instructor is paid enough for the subsequent legal training flight and the student pays an appropriate ammount per hour for legal training, the cost of the "free" leg in the middle can be made up.
Regards,
DFC