Cast when Damage Taken Support + Walk the Aether(Trickster)

The Shadow's Ascendancy class Trickster appealed to me in a few ways, but one passive node caught my eye - Walk the Aether: "Movement Skills cost no Mana" and "20% increased Attack and Cast Speed if you've used a Movement Skill Recently"

Let's say I connected a Cast when Damage Taken Support skill gem to a movement skill like Lightning Warp, if I take damage and the spell goes off to teleport me away (randomly), would I get the 20% attack and cast speed?

Thanks for looking!

Phaserun might be a better skill