The LAFD does routinely (usually several times a day) do IFTs from ER to ER and characterizes them as IFTs in the dispatch code. The hospital systems in Los Angeles, especially Kaiser, will call 911 to facilitate time sensitive transfers such as STEMIs, walk-in traumas, and patients requiring neurointensive care. This is an agreed upon policy between the hospitals and LAFD that if there is a time sensitive diagnosis and the ETA of a CCT private resource is above a certain level (I am not sure what that is) they proceed to call 911. This sends the nearest ALS engine/light force, ALS ambulance, and EMS captain to facilitate the transfer. Also, calls from UCs, SNFs, outpatient clinics are not coded as IFTs.
Also, the Orange County Fire Authority does IFTs fairly often too and, from what I understand, it operates in a similar model to the LAFD (long ETA for a CCT rig? Call 911). However, OCFA will just send the nearest paramedic engine or truck and whatever private ambulance company is appropriate for the area (Care or Emergency).
Both the LAFD and OCFA have even occasionally does emergent IFTs out of one ICU to another when there is a lack of CCT resources.