This is usually handled by the nice folks in the EMD. They explain that, per FDNY policy, we don't do transfers from an ER unless requested by the ER crew through the EMD. The FDNY attitude is, the patient is at an ER already.
Personally, I have had some call from a block away from the ER, where the patient has removed themselves from the waiting room, and request going to another ER, due to real or perceived delay in being seen. Our field-side stock answer, as my district has 2 hospitals roughly 7 minutes away from each other, is, "You want to go to the other hospital, where they have just as large a wait to be seen, and start over?" We'll still take them, if they insist.
On FDNY EMS Queens East frequency, we have a few famous frequent fliers. We had one that just clicked along, where the ambulance was en-route to him before he got off the phone with the EMD, on scene, evaluated, transported, and arrived at the ER, 2 blocks from the patent's residence, all in under a half hour. HOWEVER...a half hour after the team got him to the ER, EMD got another call from the man, who had walked home, and called again, because he felt he hadn't been seen in a timely enough manner!
Side-note on that patient: One ambulance crewman actually had a call report book of 25 sheets pre-filled with the man's name, address, and past medical history. All that was needed was the date, times, crew's badge numbers, and the current vital signs.