Yo dude.
I use Flash for that too.
Individual gifs from Graphics Gale:

Composited gifs from Flash:

but when I export/publish it always only shows the first frame, I have tried messing about the the fig publish setting making sure its set to non-static exporting.
Use this rather than Flash's Export feature:
http://swfrenderer.kurst.co.uk/Works great and saved my ass when I first found it.
Exporting from Flash will not include blend modes and other stuff in the images it spits out.
But SWF Renderer exports exactly what the swf looks like.
Get it.
on googling a bit more I see that in order for flash to export the frames the frames have to be on the main timeline
For a specific setup maybe, but naw don't do this.
Import each gif into the Library and it will automatically create a symbol for it.
Rename the symbol and shove all the images into a trash folder because they don't matter.
Arrange your stuff on the Stage and test the swf until you're happy.
Then use SWF Exporter to generate the images.
What might be happening is you have each animation as a Graphic rather than a Movie Clip.
Graphic objects need their parent timeline to have frames otherwise they wont play in a swf.
Movie Clips will play regardless of the number of frames in a parent timeline.
But I think Movie Clips don't export to images. --> most likely your problem
You have to use a Graphic instead so make sure your main timeline has frames.
That's just a guess and some Flash blah blah.
I haven't used the Flash Export option for ages cuz it's garbage and I ran into all the same problems you are having.
Don't really remember how it reacts and I'm not gonna test it.
Get SWF Renderer noted above and it will fix your problems.
I did also try to select all the frames of one sprite on the main timeline then attempt to drag the sprite, but despite me having selected all the frames in the timeline when i move the image/gif it only moves the first frame position.
Shift selecting frames doesn't let you edit object positions on the stage all at once.
That only allows you to edit the frames in the timeline.
For this you have to turn on Edit Multiple Frames. --> tiny button underneath the timeline
A range selection slider will appear above the timeline that you can drag left and right to include or exclude frames.
Then drag or shift select the objects you want to manipulate and do stuff.

If you ever need more help with Flash I've used it for a long time and am using it pretty much everyday for pixel work and compositing.
So I can prob answer any weird questions you have.
Flash is quirky as fuck and sometimes what to do is not obvious.
Despite that, I find that it's amazing for compositing loops and building asset driven pixel animations.