I am not FDNY, but this is my understanding from listening to the scanner, reading the rundowns on this forum 👍and reading the FDNY Ops Reference Manual by James Griffiths. By all means, someone correct me if I'm way off the mark.
There is a Citywide Tour Commander who is assigned for a 24 shift. This position is assigned to a DAC or AC (staff chiefs) 365 days a year. It is true that during normal business hours the Borough Commander for whichever borough the fire is in responds in place of the tour commander. Many times the borough commanders will respond on the 2nd Alarm, as it is "their yard" and perhaps, they want a reason to get out of the office ☺. However, they are not assigned until the 3rd Alarm if they did not respond earlier. Here's where it gets tricky. The Borough Commander for the Bronx may be on vacation or out sick. If a 3rd Alarm is transmitted in the Bronx, then it falls upon the assigned Citywide Tour Commander to respond. The tour commander may be the Brooklyn Borough Commander and he will respond. But there are a lot of DAC's & AC's on during normal business hours, so one or more chiefs may respond from a closer location (the fire in the Bronx may have the Queens Borough Commander or the Chief of the Fire Academy show up first). However, this would not absolve the tour commander from responding. He will still respond because the other chiefs may want to get back to work after the PWH is given and leave. At that point, the tour commander would be in charge until he saw fit to turn it over to the Deputy Chief (Division Chief), perhaps after the Under Control is given and assuming there are no other critical situations remaining that may be more appropriate for the tour commander to handle. "This is the mayor's neighborhood and he's coming up to show his presence." Of course the Deputy Chief could handle it, but the tour commander (a staff chief) may want to remain on location to meet the mayor. His call.
A five-alarm fire during the day may have 5 or 6 staff chiefs on location for various reasons. Also a 10-66 transmitted virtually anytime of any day including weekends will bring multiple chiefs, from home, to the scene. In my opinion, if I were a firefighter trapped in a building and was rescued, I would feel grateful to see 5 or 6 Staff Chiefs outside at 3 AM on a cold February morning. Why do they do it? Because they care, they are professionals and they have pride!!!