If you're looking for attacking while in motion, look no further than Mercedes, with her Piercfing Rush to stay ahead and above, Leaf Tornado to keep up the momentum in the air, great innate speed, instant ladder ascent, and a decently propelling second jump. The Nova classes are pretty mobile since they can traverse platforms with ease (Grappling Hook), they've got a second jump that takes them pretty far pretty fast (and Angelic Buster gets two). Kaiser takes them pretty far, Angelic Buster takes her the furthest out of every class, only matched by Xenon, who also goes pretty far and has a second jump. The Demon classes aren't too shabby either since they can glide and they have a decent second jump, but they also have a high jump.
Oh, youth... guided by the servant of the goddess... Unite earth and sky, and bring light to the land.