In my area, EMS is all Fire based (all local firefighters are at least EMTs and a large percentage are Paramedics) so it's completely normal for fire to handle EMS mutual/automatic aid coverage...however it's also a big no-no here to self dispatch to somebody else's call, especially out of district. It's one thing to go over the radio and say "Dispatch, Engine 21, we're closer to that call than Engine 22, we can handle it" and then let dispatch reassign you. But if the first in units need additional resources (truck for extrication, ambulances, paramedic units, PD, etc) for a big TC than it's expected that they call dispatch who can send the appropriate units and shuffle others for district coverage as necessary, so that way they're not trying to tone out Engine 21 for a call when 21 decided the radio report sounded bad and just drove down to help out 22.