Here is what works for me: while holding the mouse button down, drag the cursor into the compose pane, then outside the Firefox window, and then reenter slowly from the bottom part of the compose pane. Once the cursor turns into a hand with a "+" sign, you can drop the file(s). This way it works 99% of the time.
It usually doesn't work if you reenter from the sides, or from above, or from the bottom part of the window but outside the compose pane.
You can also try directly entering the Firefox window from the bottom of the compose pane without doing the exit/enter voodoo mentioned above, but it works maybe 50% of the time.
Super weird, I know...