New York City has carried this out one step further with the creation of the Citywide Incident Management System (CIMS). CIMS is compliant with NIMS and ICS, being fine-tuned for the needs of NYC. CIMS clearly states in a matrix which is the lead agency for the different types of emergencies that they respond to.
The command of some incidents will change as the event progresses. In a Haz-Mat incident, the NYPD is in charge until a determination is made that it is not of criminal or terrorist origin. Once that determination is made, command is transferred to FDNY.
Since the Federal government has mandated the use of NIMS in order to receive federal preparedness assistance funding for states, territories, local jurisdictions and tribal entities, it is pretty clear everyone needs to be NIMS compliant.