in the MR team i'm also a member of, we had a talk from local FB. He was saying that once FB turn up on site , they have a responsibity for Health and Safety of everyone there, and anyone going in to the cas site (or inner cordon) would have to go through their control point.
now I can think of many situations where the FB might have been called out first, and they would be totally out of their depth, eg crashed Paraglider halfway down MamTor face, or most cave rescue.
in practice i have done this a couple of times, as stated before FB got called out by accident, and it wasn't a problem.
however if you take it to the letter, they may beleive that they are in control, even if they have no knowledge of caving or of that particular system.
In reality i think they would readily concede.
they do have resources that may be able to help (eg high volume pumps), and we have called them in to back us up with concrete chainsaws for a stuck weaseller.