I would much rather them work like the equipment instead....
Granted, tbh tying them to the MC one way or other (as equipment there is also no easy way to switch between MC and another) is technically unfair in term of main char choices. It is best suited for fire and lightning which mostly only have one role--damage. But the other three have alternative roles (tank, healer and support(dance)) which might be used for normal team but swap for damage during for fox etc.

I propose we tie the gate runes to the SKILL order in the skill box thing. Because the skill order does NOT actually affect the battles themselves, so you can swap them freely and easily. This makes changing char much easier.