Farming old raids and dungeons has always been a popular activity for many players because it allows you to have a shot at obtaining rare stuff, pets, mounts, achievements or w/e.
And since Blizzard has nerfed mobility for almost every class in Legion, running old content has become slower and less comfortable.
Now, while demon hunters shine in terms of getting around, I would say that they can be rivaled by outlaw rogues who I believe can achieve the maximum permanent movement speed on foot.
Base movement speed: 100%.
+15% speed: [Fleet Footed] passive ability (learned at level 58 )
+15% speed: [Hit and Run] talent (unlocked at level 30)
+10% speed: [Minor Speed] enchant (cannot be applied to items higher than ilvl 600):
+2% speed: [Gift of Versatility] enchant (can be applied to a cloak of any ilvl):
Total: 142% passive speed.
Now, why outlaw?
You're obviously going to use The Dreadblades as your main weapons, those blades have a hidden ability [Taste of Blood] which increases your speed by 2% after killing a humanoid enemy. While 2% might seem a minor increase, this buff stacks to 10, so after killing 10 humanoids you'll run 20% faster. Lasts 30 seconds.
So... 142% passive + fully stacked [Taste of Blood] = 162% speed.
More than that, [Bear Tartare] food buff increases your speed by 70% for 5 seconds after killing any enemy (credits to advanta for pointing that out!).
So with all those buffs applied you can run at 232% speed.
You can also sprint on a 1 minute cooldown, however it won't increase your speed beyond 232%. Use when [Taste of Blood] is absent.
It's possible to achieve 147% passive speed if you go assassination and pick the [Nightstalker] talent (only works while stealthed though).
Some WoD and Legion items also have a passive speed increase as a tertiary stat, I can confirm it can increase the speed you run at even further (got an item with +559 speed which increases my running speed by 6%, resulting in 148% passive).
List of items/enhancements which DON'T increase speed any further:
[Figurine - Golden Hare]
[Defiler's Leather Boots]
[Swift Skyflare Diamond] and all this kind of gems.
Slightly offtopic:
Handy macro which allows you to spam one button and destroy big packs of lower level mobs as an outlaw rogue (turn on Blade Flurry first):
/cleartarget [dead]
/targetenemy [noharm, nodead]
/castsequence reset=2 Marked For Death, Run Through
And since people tend to argue unless they're provided with a proof, here's a video.