I've already mentioned the script that I use to scrape the browser's cache for video files. But the real solution for off-line viewing is to download, or record, the streams to files.
One easy way of doing this is clive. It works with several video sites - I use it regularly to download video clips from youtube.com and google video. Highly recommended.
Some video clips, however, are more elusive. The major local news sites around here serve mms:// and rtsp:// streams (typically accessed via URLs linking to files with .asx extension). These streams can usually be recorded with MPlayer, like this:
mplayer -dumpstream -dumpfile video.wmv -nosound -vo null mms://www.example.com/path/to/video.wmvor
mplayer -dumpstream -dumpfile video.wmv -nosound -vo null -playlist http://www.example.com/path/to/video.asx
The stream URL can be found in several ways, e.g. by inspecting the source code of the web page serving that video stream; or by sniffing out the URLs being accessed by the web browser.
What I am still looking for is a way to dump rtmp streams. If you know how to do it - please leave a comment!