Where are FLV cache files stored?

This may vary by browser, but in Safari, if you press ⌘⌥A, you bring up the Activity window. Find the largest file being downloaded for the current page; that is generally the Flash Video. Double click it, and it will be downloaded to your downloads folder.

This avoids having to learn the various places Mac OS X will store temporary files on disk. In general, things that aren't expected to be cached for re-use end up in /private/var/tmp and things that are in ~/Library/Caches but in your case, safari will expose that file more directly and allow you to fetch it.

If you are ever curious - you can watch filesystem activity in real time with some quick commands like sudo fs_usage -w | grep Safari

As far as I can see, using Firefox on 10.7.3, the flash video gets stored as a file called FlashTmp.XXXXXX, where the Xs are randomly generated.

This file is in /private/var/folders/, 3 levels deep in randomly generated directory names like ax/4fis_mf2993_23iofdasiof29309sfd/J.

I found it by using the command lsof|grep Flash.