Does it work when adding to mx.flash.UIMovieClip rather than UIComponent?
Do you know how big the movieclips need to be?
No, it doesn't
I am building an application that is totally design in Flash.
I have a lot of movieclips, each one in a different size.
Have you looked into using the Flex Component Kit for Flash?
If you want to use components for Flex 4 using Flash CS4+ you will need to use the latest MXP file here:
It's not quite clear (at least to me) what you are trying to achieve.
What kind of functionality/interactivity you want build in your Flex application besides laying out the movie clips?
If it's just the ability to load them dynamically and trigger them, then you could use SWFLoader() and within the script block listen to the event - such as mouse click for example - to start playing particular movie clip: SWFLoader.content.addEventListener(...)
If you want your movie clip to communicate/interact with your main application in more elaborate way, then you need to create custom event and implement such class in both - the MovieClip that you make in Flash and in the main application