What are some champions that can use their abilities while walking?
i can only think of sona and syndra
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I clicked on this thread to say 
lol
Not including mobility spells (or on-hit auto reset buffs)
E
W (recently changed to :D)
E
QWER
W ( I THINK he can fire rockets without interrupting movement, not 100% sure)
QWER (not 100%, but fairly certain you can taunt while running past an enemy in powerball for a "drive by taunt")
Q (possibly her W, not certain about that)
QWE
E
QW
QE
WER
R
W
R
E (the second cast of E, cripple, does not interrupt movement, the tempest ground smash does have an animation)
W
most spin to win champions such as
QW
QWE
WE
WR
W
ER
E
most buff/self-buff champions such as
W
E
W
W
WER
W
W
W
E
W
E
Not certain, but I think
Q
W
W
E
Toggled or abilities that charge up(requiring two button casts) such as
Q
Q
E
R
W
Q
Have not covered all of them, but here's a bunch
Ahri?
Can use his W and can use her Q
Hm...
Eve can use her Q and W. Nid's kitty W. Gnar's E. Jayce's E in cannon stance (I believe, don't quote me, been a while since I played Jayce)....
I can't think of any more. xD
, her Q. She can throw in spot and run to catch it in a different position.