If it is a frame by frame animation try importing the MOV file into Flash rather than exporting from AE.
Not sure if this will work but it might. When you publish the file in Flash you will get a resulting SWF file that is created automatically. See if it is smaller.
Are you using effects ? because if you're just animating your AI imported shapes, AE should export a lightweight swf.
I am not using any effects, but the seemingly small animation is 1.2 MB when I select Export>Adobe Flash Player.
The AI files which make up the animation have a few small gradients in them. Is that what's making the files so huge?
Are there some special setting that I should select to make the file smaller?
(Also worth noting... I tried to select "Ignore" instead of "Rasterize" but much of the objects disappeared from the final animation.)
Did you read the list of unsupported features that cause rasterization that I linked to?
> The AI files which make up the animation have a few small gradients in them. Is that what's making the files so huge?<