It is tempting to use the advantages of Flash mature animation tools, but unfortunately there is no "out of the box" solution.
There are however ways to get the job done:
1) Use tools like CreateJS to export the animation as a JS file. Then make a script that will parse the data to a *.plist file with links to exported images, their position etc. Later use a tweener to animate the graphics.
You will also need to manually draw the vector shapes and save them as .PNG files as CreateJS doesn't export them as PNGs. But it's easy since CreateJS uses the save API as Flash and you can get all the shapes accurately.
The downsides are:
CreateJS has no support for complex gradients
Parsing JS to PLIST is a pain, it took me a whole day to write it. Be ready to use a lot of RegExp
NOTE: if you'll use CreateJS toolkit in the settings uncheck the Compact shapes option, it will make vector data to appear readable.
2) Make a tool that will read the SWF with animation and save each animation as a PNG sequence or a sprite sheet (sprite sheets work better with COCOS2dx).
The downsides are:
- File size will grow comparing to SWF because you'll have every frame of every animation as raster (unless you have vectors so complex that they take more space than raster).
- You'll have to have all the animation present on the stage, because it's problematic to target MovieClips from the library, or impossible if they're not exported for AS.
3) Dive into JS:FL and make your own tool that will export all the graphics in a format usable to you. There are no downsides, but I find it harder to do.
4) If you don't have multi-level animations with MovieClips inside MovieClips you can just right-click on an animation and export it as a sheet/sequence, it's very fast to do, but you'll spend a lot of time editing FLA file to meet the requirements if you have multiple levels (you'll need to convert parent MC to a graphics so it would play on a single timeline).
You may want to export the animation in alternative ways rather than sprite sheet or sequence, like a bone animation where you save every part of a more complex animation and a list of tweens/animation points.
I wrote a about it here: Using SWF animation in Haxe/OpenFL applications It's not about COCOS, but it will work with any technology universally of platform.
As you see it's not all bad news, but you'll have to do some coding for that as there are no public tools that do that. I had to make all of the tools listed above and I can recommend you to stick with the 1st choice.