When a party is not fighting or interacting, just walking around a city or exploring a dungeon, do they need to respect their walking speed and take turns or it's ok to walk freely?
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You would use turn-based movement only for tactical movement, e.g. combat encounters. Other than that, you can move freely (exploration would be local movement in D&D terms, measured in ft/min) but you should roughly keep track of the time (if you explore a large dungeon, it would take much longer than for the cellar below a farmers hut).
Exploring anything larger than a few grid tiles in turns would be beyond boring.
The real answer is "it depends".
If you are in town and want to head from the tavern to the blacksmith to get your sword fixed, then no you shouldn't need to walk according to your move speed.
However, if you are walking along a narrow ledge or over a bit of difficult terrain, it is fair for the DM to want you to move things tactically to either be able to know where precisely you are walking to know if your character steps on a weak spot that will fall through... or to just make the PCs more paranoid.
It's fairly traditional to ask about a party marching order, both to establish what light sources are being used and also to determine who's going to notice when the monster pops up, the ground crumbles away, etc. That can weigh into whether you need them to establish turns and the like.
One way to handle it is to just have them make Search rolls as necessary, whether just at the points of interest or frequently to keep them vigilant (this requires some knowledge of how your party works. Sometimes, this builds immersion and sometimes it results in the party stopping every 10 feet to search the corners and poke the statuary with a 10-foot pole "because the GM just described it and that means this must be important") and otherwise just treat it as "time passes, now you're at this significant point where you need to make a decision".
Another is to simply let the players designate people scouting ahead, sticking with the main party, etc, and make their rolls and have those people as the ones who are likely to run into the initial stages of trouble. If everyone's searching at the same time, it usually works more like the first case. If the rogue or the ranger is scouting ahead, it might mean that they have a round or two of combat before the rest of the party arrives, which can nicely build dramatic tension as well as encouraging said front-liners to be smart about when to fight and when to retreat back to the main party.
Lastly, you can run the whole thing step by step. This is probably too detail-oriented for most groups unless they're in a situation where they are in imminent danger and time is important to track, such as exploring a dungeon while being pursued by a raiding party. The moment-by-moment treatment emphasizes the time pressure.