Tom posted on a great webapp VIXY that captures flash-based video. I have used this app a dozen times since he showed it to me. I found a variation the theme with Hey!Watch. Hey!Watch lets users upload video and encode it for devices that range from iPods and Zunes to NintendoDS and PocketPCs. It also has a list of formats that will accommodate most of your needs.
You are free to transfer video from flash-based sites like YouTube, as well. They provide additional options if you subscribe, but the free service seems worth exploring. If I am reading the site info correctly, they will even let you automatically transfer video from an established podcast. The interface is simple enough that your average tech-minded teacher could upload raw video and have it converted to a format for display/distribution with ease.