I took that time to actually uninstall Firefox and reinstall it cleanly. Over the past year or so, Firefox had gotten more and more sluggish, especially when downloading files.
As soon as I clicked a link for a download, Firefox would become unresponsive for at least 30 seconds or so. And only after that, it would start downloading.

Of course, reinstalling Firefox meant I lost quite a few settings (like saved passwords), but I’ll live with that. It’s not like I don’t actually KNOW those passwords .. it’s more just convenience.

So after deinstalling, removing the firefox directory from Program Files, removing the firefox directories from my profile, I reinstalled.
Then I hunted down my favorite Firefox addons:

• Adblock Plus
• NoScript
• Gmail Manager
• CustomizeGoogle
• Tabbrowser Preferences

On top of that, I installed the Littlefox theme, for more screen space.

Now downloading works the way it should again. Immediate response when clicking a file, instant start of downloading.