This snippet is from the .pdf file from DJI for updating firmware:
Updating Intelligent Flight Battery Firmware
The Intelligent Flight Battery is upgraded during the aircraft firmware upgrade process. It is recommended to keep the upgrade package files in your SD card. The upgrade will start automatically after power cycling the aircraft.
That certainly sounds to me like - when you get a new battery, the firmware update must be "applied" to the new battery.
I've kept the latest firmware update on my SD card - just the .bin file and nothing else. Guess we won't know for sure what really happens until someone gets more batteries!
Upgrade package file... this would be the zip file and the P3 well ignore this file. It's not the bin file. If the bin file is left in the root of the SD card the P3 will attempt to update each time it's turned on.
As I understand it, the firmware which updates the P3 also updates the battery. However, the bin file needs to be on the SD card and the P3 needs to be booted up with each battery so that each better is loaded with the new firmware. After that, remove the bin file.
There is a possibility that the P3 itself will update each battery as they are inserted and that the bin file is not needed on the SD card for this but I doubt that is how it works. I also think DJI will tell us when I new firmware will be updating the batteries.