What are Flash videos?
To put it simply, there are two kinds of Flash-based videos: traditional Flash videos (also having the .swf extension) and enhanced videos having the .flv extension. The former isn’t really suited for Web-based distribution (it, for example, doesn’t support server-supported location change without fetching the entire clip first). The latter is almost exclusively used by large media distribution sites like Google Video or YouTube because of its technical superiority.
The Pocket PC and Flash videos
While .swf video files are played by almost all Pocket PC-based SWF players as can also be see for example in the article Playing Flash Content on the Pocket PC, playing the latter files on the Pocket PC is a lot more problematic.
Much as some people say the latest (version 7) Macromedia Player for Pocket PC, this will not help with the majority of the “production” Flash videos out there. Most of the videos (for example, the Google Video ones; the YouTube videos are said to be Flash 8+-only) are playable neither from remote locations nor from the local file system.
Note that NetFront 3.2/3.3 isn’t able to play these videos either as can be seen in this screenshot. Renaming them to SWF doesn’t help either – as is the case with the Adobe plug-in and PIE/IEM.
(For testing purposes, I’ve visited the Google Video archives (for example, this Shakira - Hips Don't Lie spoof video).)
Convert to be able to play!
You can, however, convert these files to a format digestible by , say, TCPMP, the best Pocket PC-based multimedia player. To do this, download for example Total Video Converter. Download, install and start it; drop the input FLV file in the “Task list” list (it’s in the upper left region), choose MPEG4 Video in the Mobile group and click the Convert Now button; the conversion will start. The output file will be created in the C:\Program Files\Total Video Converter\Converted directory. Note that you’ll also need the AAC codec for the sound of these files to be played; please read this thread on getting and installing it. If you don't want to install it, go for another output format.
Alternatively, you may want to use the free (!) Mencoder as is described here.
Note that some people also recommend the free Super by eRightSoft. Some other people, however, say it's causing synchronization problems.
Finally, you may also want to give a try to the commercial Replay Converter by applian technologies. The homepage of the developer is also worth checking out because they are also offering Replay A/V, which can also help in downloading FLV files (in addition to, say, YouTube, Google Video etc. videos) directly off the Web, in case you don't want to manually look for videos in the browser cache.
Also note that you’ll want to read Playing Flash Content on the Pocket PC on the secrets of acquiring FLV files if they are residing on a Web server like Google Video (if you don't want to go for the above-linked Replay A/V).
Also note that the quality of the resulting video file will be really inferior to that of the source.
Other discussions of this question
Recommended generic article on Flash & PPC
Playing Flash Content on the Pocket PC
UPDATE (10/24/2006): there is a brand new, free service here allowing for online FLI -> MP4 / 3GP conversions, which, then, can be played with both TCPMP and CorePlayer. Please also see the comments section in here for more info.
I also recommend the following discussions of this article: AximSite; PPCT. I've also elaborated on whether alternate Pocket PC browsers are able to play YouTube videos here. A worthy PPCT discussion of the latter article can be found here.