You enter turn-based action when you need to track time that closely. You leave it as soon as you don't need that close tracking.*
Turn-based time tracking dominates combat, but it can also be used for chases (DMG 252), complex traps (DMG 121), even tense social scenes. That said, it's a really clunky mechanism, right up there with alignment and "what HPs mean" for likelihood of starting an argument.
You're right: while the books say that initiative is rolled at the start of every combat encounter**, they say nothing about when to leave initiative. Common sense has to be our guide: we clearly don't want to always be in "combat-time," or we'd never get from one side of town to the other. Given that we will sometimes not be in combat-time, the question is then: when? Whenever you don't need it--initiative, and combat turns/rounds, are a tool for you to use or not as you need.
A few observations, from my experience playing and running:
- initiative variants can also play strangely with "roll for initiative"-based features--be thoughtful when employing them;
- I don't even use initiative for everevery combat encounter--sometimes it's not worth the trouble (in my opinion) and I'll suggest to the table that we run a simple little simultaneous-declaration-and-resolution combat;
- escaping combat while in turn-order is largely broken in 5e, in my opinion--the moment either side decides that "running for my life" is more important than "exerting my will by force of arms and might of magic" I break out of combat spacetime.
* - To the extent that initiative and turns/rounds even do that.
** - Which can be a little clunky, like a voice from the gods screaming "MURDER NOW!" To fix this I ask players for a set of initiative rolls each time we leave turn-order, to have on hand the next time it arises.