23rd) For detailed information on Air Port base station firmware updates, including instructions how to install the update, please visit:

I suspect the process is essentially the same on an Apple Time Capsule router but I don’t have one with which to verify these instructions.
The screens below show the upgrade from Firmware version 7.7.3 to 7.7.7 released in mid-2016.
For the Open Air Port Utility on Mac Display badge for connected Air Port device in your network if firmware available for update. Note: Downloaded Air Port Firmware saved on Mac, ~/library/Application Support/Apple/Airport/Firmware.
When you open Airport Utility on Mac from Applications/Utilities folder, Airport utility will display badge to the right. Sure about you are connected with internet, Open Airport Utility App from i Phone/ i Pad.
Apple’s release notes says that the update fixes an issue that prevented Air Port base stations from appearing in the Air Port Utility when Back to My Mac was enabled.