I've checked both of the suggestions and now realize that I omitted an important detail: When I used Zamzar, I would upload a video by pasting its URL, then Zamzar would email me a URL for the download page where I would go to obtain the converted file.

I really can't recommend one. The only place I ever download streaming video from is Youtube, and not often at that. I linked the extension I use for that here but it only works on youtube.com. What I would probably do is go into "Add-ons Manager" and search for something like "video downloader" and look though them for one that purports to work on the sites I intended to use it. I'd also look at the number of users as well as the rating, with the realization that nothing works for everyone. Perhaps somebody else will come along and recommend one from personal experience, I'm sorry, I can not.

I'm currently using "Download YouTube Videos as MP4". I don't use it often but it seems to work fine.

as always I suggest going to gizmo's site for the best recommendations Best Free Streaming Media Recorder