Hi everyone. 3rd Edition had a tendency to have static combats. Full-Attack Actions made it so people didn't want to move. 4th Edition has a lot of powers with movement built in, and I felt there was more movement then; defenders/soldiers locked people in, but they weren't in every fight. 5E doesn't have full-round actions, but my experience has been that fights are more static than 4th but more mobile than 3rd.
Nod. 5e doesn't have full attacks or full-round casting times (or full-round anythings, really), so if you don't have to choose between moving and acting - but, it also has relatively little reason to move, few forced-movement effects, and little benefit to positioning (and, run TotM, it encourages you to ignore/fudge positioning).
So, your experience seems consistent with the presentations & content of three rulesets in question.
Today, I started to question if universal attacks of opportunity were the problem.
5e OAs are pretty meh - they consume your precious reaction, and they aren't provoked by much (particularly, not by ranged attacks or casting in melee, so the kind of gank-the-caster/protect-the-caster dynamics you'd sometimes get going back to the early days, is less noticeable), and they don't do anything beyond single-attack damage, which scales little compared to Extra Attack, cantrips & spells.
What have your experiences been?
I've found 5e to be functionally as 'static' as 3e. There's more movement, but it's very often in the form of "I move up to the next guy and attack," adding nothing much to the encounter - just part of the focus-fire routine.
I've seen DMs adapt 3.5 or 4e grid use wholesale with fair results. Still not a lot of movement, though, outside of Cunning Action and similar monster abilities.
Special attacks that also allow you to move is the solution
Ordinary attacks already also allow you to move.
I find that removing the boardgame aspect (that is, miniatures)
You mean the wargaming aspect, then. ;P
tends to get people to be more descriptive with how their characters are moving and interacting in combat.
I find the opposite. Put out any sort of visual aid, and players will focus more on positioning. In the absence of that, you get - in addition to a lot more time spent describing and re-re-re-describing the scene and how it's changed since the last turn to every player, every turn - a lot of very meh declarations (keep attack the same one, attack the nearest/next one, etc). Blah as can be. Doesn't matter how evocatively you describe the area, either.
(Yeah, go ahead, insult my players, you know you want to...)
Of course, what can get players more descriptive in TotM is the carrot - if you often (but not too consistently) hand out advantage or lesser bonuses for such descriptions.
Also, IME 5E monsters drop far more quickly than either edition, so there is quite a bit of movement to face new enemies.
There is that, too. The 'fast combat' goal is antithetical to the development of more dynamic combats.
Since you only take OA when you leave the enemies reach, you can move around enemies to get into a better position (such as to attack a different enemy).
Yep, it's like everyone has the 'Pass Forward' fighter utility from 4e (which was a low-level, at-will, because it wasn't really super-useful, though I quite liked it in one or two builds). Unless you're tracking positing very carefully (and have added some rules to make it more important), though, it'll rarely make a difference.