I think mobility, like most other mechanics, can be healthy and fun when applied in proper context. However, the problem with mobility in League is that it has become practically mandatory to kits, and as a result has become limiting. When a feature is repeated so often in kits, it feels like that feature is more of a limitation, and when too many of these mandatory mechanics pile up, the results often end up feeling like the same collection of features rearranged in slightly different combinations. Percent health damage is another example of this, and has been present on almost every new or reworked kit in recent times.
Mobility specifically, though, has caused a great deal many problems of its own: the first is that, when it offers players a way of quickly closing the distance to reach an opponent, it effectively renders battlefields smaller by comparison, since much of an area can be bypassed. League's map hasn't really evolved with this, which has combined with recent damage increases to create a Call of Duty-like atmosphere, where an opponent on the edge of your screen can catch up to you in a couple of seconds and kill you without giving all that much of a reaction time. Additionally, when these dashes are offensive in nature, they also tend to render games more snowbally, since those dashes become super good when you have an edge on an enemy (you get to kill them more), but useless if you're behind (dashing means offering yourself up to get killed). Because most dashes bypass terrain, this also tends to trivialize terrain from both the map and champion kits, and again causes the map to age poorly. This is likely why Riot has experimented much more recently with giving champions dashes that don't cross over terrain, namely with Ornn, Urgot and Galio.
Overall, one of the problems with mobility in League is that Riot has always had some trouble estimating its power. Originally they completely underestimated the strength of dashes, which led to waves of mobile champions becoming OP, but even now it's not always clear if mobility is being used with a specific purpose, other than to make a champion more fun to play: the problem with giving a champion "fun" mechanics purely for the sake of fun is that, if those mechanics don't relate to their theme or kit, the champion ends up becoming shallower and less consistent. If the champion ends up being designed around the mechanic , and if that mechanic is too present, kits end up feeling too similar, a problem that exists with several classes (divers and assassins have a lot of overlap, for example). Moreover, designing champions around "fun" mechanics purely because those mechanics are popular, as is the case with dashes, ends up falling into a trap, since while those champions can generate short-term interest, in the long term they lose value if not counterbalanced with alternatives. As a result, because Riot has tried so hard to capitalize on mobility due to its popularity, League has ended up losing a fair bit of diversity, since having mobility is no longer something special.
Because of this, I think the solution should be to desaturate League of its excessively repeated mechanics, which among many others includes mobility. In many cases, certain champions either have mobility when they shouldn't, or simply have too much mobility. However, the fact that the near-totality of melee champions have some form of movement enhancement to me suggests that mobility is also somewhat of a necessity: Gangplank, for example, has a 30% movespeed bonus that is practically always-on in fights, because that's how much he needs to be able to move around properly. By contrast, Nasus, despite all of his terrifying power at melee range, and tremendous durability, suffers in many teamfights, because his total lack of mobility tools means he can get kited ad infinitum. It might therefore be the case that melee champions are given mobility out of necessity, and not just as a fun addition, which is one reason why their kits come across as overloaded. In this respect, buffing the base movespeed of melee champions, and removing some of their extraneous mobility could therefore allow melee kits to be designed without always having to resort to gapclosers or hastes.
could counter Darius ult? And 
is "mobile" just because he has an enemy reliant point click dash that can't be used in any other way or freely to escape. yeah, you're right, Wukong is a mobile champion.
relies on other champions/minions/monsters to be mobile. I don't think they have true mobility. 
aren't mobile. Having a dash/blink doesn't make champion instantly mobile. Champions like 
to be immobile and 
If you would genuinely consider TF a mobile champion then I'd hate to see the stone walls you consider immobile.
did but i know she can with parry now
