Well, I've had a response from those in the know and the answer is that training flight requires the FI/CRI to be captain. So he must be qualified on type, including having received the relevant differences training or having grandfather rights.
The idea of swapping captaincy in the air is ill advised at the best of times. Taken to the absurd conclusion, this would allow some young C152 FI to conduct the training flying on something with VP, RU and TW as long as the gear was up and the prop set. Then return captaincy before the prop was adjusted and the gear lever moved.
Keep it simple, keep it safe. FI/CRI conducting training flights are PIC throughout and must be qualified on type.