I always surmised that the period between V and VI was essentially a "peaceful" period for the Federation and Starfleet. Perhaps due to the crew of the Enterprise and Captain Klaa's Bird-of-Prey. Oh of course there could have been other "skirmishes", but not major ones.
Starfleet perhaps was able to spend more time on exploring and scientific studies. Which explains the reason that both the Excelsior AND the Enterprise were carrying probes for such things as "gaseous anomolies"..(insert your own joke here...
Hence I believe that Kirk was still in command of the Enterprise with mostly the same bridge crew, except when Sulu was promoted and given the Excelsior. Perhaps with all these boring exploration missions is when everyone decided to retire.