Even with over 400 move speed her animation is still the slow walking one. I just tried with winter lulu and noticed it on star guardian while playing.
You only see the running animation with homeguard effect at the start.
As a Lulu main myself, this bothers me greatly, as I hate her default walking animation. I tried with every single skin, to no avail. It seems when you gain a large amount of movement speed, her running animation triggers. (Ascension boost pads, Homeguard, ect.)
But c'mon! This is ridiculous. I agree it's not one of League's bigger problems, but it's still very annoying nonetheless.
