A client wants me to do test animation with his already animated flash file (multiple characters, backgrounds etc). I can use any software I want, and I want to do the project in after effects, simply because I'm more comfortable using it. Is there a way to export his assets out of Flash and import it in to After Effects, so that I can rig and animate them in AE?
AE can only use published SWF files and with serious limitations at that. Otherwise Flash supports exporting stuff as images, AI files and SVG, but naturally even that will require specific preparations and the exported files extra work in Photoshop and Illustrator. I'm afraid this is going to be a lot more complicated than you and your client probably thought.
from After Effects Help, more detail about importing SWF assets:
SWF (continuously rasterized)
SWF files are imported with an alpha channel. Audio is not retained. Interactive content and scripted animation are not retained. Animation defined by keyframes in the main, top-level movie is retained.