Both my old department and my current one do this. At my old department I was a Fire Department Ambulance Operator (AO). We were single role EMT only on the ambulance, with each Engine company being a 4 man crew with a minimum of 2 Paramedics on board. If the call was BLS transport the engine would go back in service while us EMTs on the BLS ambulance transported to the hospital. If it was an ALS transport, the Paramedics would ride in on the ambulance with the patient (the EMT attendant would be in back helping out...taking vitals, patching up the 12 lead, spiking bags, etc) and the engine would follow up to collect the medics at the hospital.
My current job is very similar in that I'm an EMT for an ambulance service that responds a BLS ambulance with ALS fire. Only instead of a fire department employee working out of the fire stations I'm a private ambulance employee, we have our own dispatch with a mix of street corner posting and our own deployment stations (fire sends our dispatch the call info and they send the closest ambulance from there whilst juggling units around between stations and posts for coverage). Otherwise we show up in a BLS ambulance for all calls, fire shows up with a 3 man Engine company and a 2 medic non transport capable squad. If it's BLS we take it same as before, and both fire units go back in service, and same as before if it's ALS the fire medics on the Squad follow up, one medic riding in back on the ambulance, one driving the squad to the hospital. Meanwhile the engine goes back in service and is thus available for first in assessment of the next call in its district.
In busy areas it actually works out fairly well. The engine can cancel the squad if they determine it'll be a BLS run before the squad ever gets there. And when we have rigs stacking up at the hospitals on busy days it keeps ALS units from being needlessly tied up on BLS patients. (Likewise if the squad is on scene first they can cancel the incoming engine if they don't need the manpower)
Like I said, it seems to work in busy areas that get 10-15+ runs a day where even being cancelled enroute a portion of the time and only transporting ALS runs and being able to clear the hospital quicker than the ambulance ALS squads can get spread pretty thin. For the less busy areas....yeah its slightly redundant lol
But in answer to the OP's original question, yes ALS Fire here can legally assessment a pt, determine them to only require BLS monitoring for transport and have a (private or not) BLS ambulance transport the patient while the fire unit goes back in service...our entire system concept kinda depends on that lol.