For the record:

The Flash plugin was not working in Firefox in both my desktop (64b)
and laptop (32b), but it was working fine in Chromium.
I finally had time to troubleshoot it and I saw in aboutlugins that
I had 2 Shockwave Flash plugins installed. One was 9.something and the
other the current 10.3 r181.

The culprit was the package 'swfdec-mozilla' which I had installed in
addition to 'flash-player' and 'flash-player-kde4'.

After deleting 'swfdec-mozilla' and 'swfdec', Flash now works fine in Firefox.