I had the same problem, temporary solved hacking the ezfilepasstroughhandler.php. I tried to use mod_headers also in apache virtualhost configuration file, but doesn't work...
The same happens with eZ 3.8.6, temporary hack for me was to move flash out of content, and into image-catalog. I have just one flash on the frontpage, so not a big problem, but i will investigate some more to make this work properly.
The problem was fixed by introducing a new enhancement. See the issue report for more details. Maybe the best solution is to backport this enhancement to your installations. Good luck!
