In this instance, Kirk and the company of the USS Enterprise Circa 2271 fell into a Swirly Thing and found themselves in the lap of the USS Kongo NCC-10455. After being declared relations of the Epiphany Trek universe versions of themselves, most of which are dead, they joined the staff of Starfleet, doing the Starfleet thing. See: Time and Again.
They we given the more modern USS ShiKhar equipped with a Johnson continua drive and tried to find their way to their home universe. Eventually Kirk and company returned and decided that close enough was good enough. They were assigned to Starbase 600 and command of a new Dolphin class ship building there.
Kirk took command of the USS Sigma Seven and is on duty to this very day, seeking out new Life and New Civilizations.
Kirk has been adopted by a bachelor house of Aneilogs in the Emerald City. They give him an anchor and a place to go when on leave. Kirk would deny up and down that he needs or wants an anchor, but he keeps going back to them.