Indeed, crew transfers had to have been somewhat common on the Enterprise since no one questioned it when Norman arrived on board and claimed to have recently transferred to the ship.
On the other hand, (except in terms of the description in STID), a five year mission could be nothing more than a reference to the fact that the ship is set to a five year term of duty before returning to spacedock for refits and upgrades, and possibly a major shake-up in the crew...
According to the original 1964 series pitch: (By the way, I'm surprised to see "Federated" there, since I thought the idea of the Federation hadn't been coined until midway through the first season. Still, it's clear that at this point Roddenberry saw the ship as an Earth vessel rather than one representing a multispecies union.) So basically it's about patrolling the frontier. There wouldn't have been much established law, order, and government out there yet, so they'd need Starfleet to be on hand to respond to distress calls or defend the borders or support colonies in need, and in between such crises they'd spend their time exploring the still largely uncharted frontier, filling in the body of knowledge. So it's basically about maintaining an ongoing Federation presence in a frontier region that doesn't yet have much formal government or support institutions in place. I don't think they ever intentionally returned to Earth in their own time. In "Tomorrow is Yesterday" they were "on a general course in [Earth's] direction" when they were flung into the past, but they weren't actually headed there. As for crew replacements, those would've come aboard when the ship docked at starbases or the like. Starbases were the equivalent of forts in the Old West, footholds in the frontier region and support bases for the forces that patrolled the frontier and protected colonists and settlements.