I want to see KyoAni actually do a show with action sequences and see if they can pull anything off or whether it would be pedestrian.
A couple have already been mentioned, but there is also The Second Raid. It's not as impressive visually as their more recent work, but it still has plenty of well animated sequences. You can dig up some of their recent commercials and teasers that really show off their action skills.
I can't wait to see Chuunibyou demo Koi ga Shitai!..
GodAni is clearly the studio with the most interesting titles.
What does that have to do with visual quality?
There were hand-to-hand combat sequences or sword fights in Nichi?
Yes, for comedic purposes of course.