Lee Brimelow has a tut for going from AE to Flash, and he explains how to keep files small, what you want is about 60% of the way through the video tut. Small files = vector files, anything rasterized (most effects) are huge. If you stick to Illustrator files (as layers in AE), solids with masks, text, and a very few effects (like Audio Waveform and Audio Spectrum) you get vectors and small swf files. Not the best of news, but this might improve in the future since Adobe acquired Flash (and Flash can use Blur (rasterlike effect) and more via the Flash Player at playtime) so maybe Adobe will make AE more Flash compatible, size-wise, but for now you stuck with the limitations listed.
https://forums.creativecow.net/cgi-bin/page_wrapper.cgi?forumid=2&page=https://www.creativecow.net/articles/brimelow_lee/ae4flash1/index.html
One interesting thing is that if you output a swf from AE (with “Ignore Unsupported Features” and get your vector small swf file), then bring it into Flash and Publish that, the same file will be substantially smaller (sometimes dramatically smaller). Evidently Flash optimizes the same file better than AE makes it originally.
As an alternative if you have lots of raster effects from AE, you might want to output your file as an flv, Flash’s video, very usable in Flash.