You can try and ask  jasonafex and see what he says! Although my interactions with him have been less than enjoyable.

I know he uses gif mostly, but you can see what  trunchbull says. He may focus on gif, but he's a very skilled animator.

If you're using Flash to animate, you can just render it as an SWF file which works with FA. Or even MPG if you want to upload it to Youtube. Other than Flash, I don't have much knowledge. :\

as was stated the SWF is the BEST way to get animation onto fa as for youtube i find wmv and or avi formats to work well

If you can export to any kind of video format, I've had pretty good luck with a video to SWF converter. It's cheesilly called ''Freemake Video Converter'' but actually does a very serviceable job all things considered.

Though, of course, trying to get anything longer and or sizeable to comply with FurAffinity's filesize restraints is going to be tough via this method. It's possible, but it'll turn the output into a muddy, pixely mess due to compression.

If you know the timing you want to use, you can just import .pngs from other programs to flash. I would suggest doing multiple layers, a background layer and then layers for the parts you want to move, to save on filesize, for FA uploading~

I'm not sure, but I do know that  FuzzAmorous Is a SUPER talented animator and might be able to help with a few helpful hints and tips if he's not busy!

Not sure what you mean. I've exported FROM flash, but you're saying exporting TO flash? So like, importing images?

Import images to flash to animate
-Import > Import to Library > Ctrl +L (opens library,) Move images into the stage (working area.)
-Make a new keyframe (F7) for a new image, and rinse repeat. Animation. :P

If you are Animating IN Flash and then export from flash as a .swf (FA friendly.)

With either option, you can come out with a .swf file easily. To do this, you would go to File> Save As, and save your project as a .Fla file.
THEN, File > Publish.

This saves a .Swf file at the same place you saved your .Fla

If you meant something else, please lemme know. I've used Flash for years. :I There are more complicated ways of exporting/rendering, but I figure this is what you were asking.

Sorry I'm late to the game but I thought I'd offer one suggestion.
I dunno if you ever use photoshop but there's an animation plugin for it called AnimDessin 2 that works pretty well.
Here's a link to the plugin http://pspanels.com/animdessin2-2/
Here's a video tutorial/description https://vimeo.com/96689934

Here's my own method, not only is it easy, it also makes file sizes very small with very little loss of quality.
http://www.furaffinity.net/journal/6991075/