The videos we put together for anyone who asks are slideshows, and even if you don’t want to go to the expense of getting someone to make these for you, we encourage you to create them yourself. They don’t just have to use static backgrounds, however. There are a number of sites offering short video clips free of charge, including blurred ones which are ideal for this purpose. The good people at Noupe have compiled a list at High-Quality Stock Videos: 20 Free (or Almost Free) Sites to Download which is well worth bookmarking.
Alternatively, you can blur your own video to use as a background in Final Cut Pro or Adobe Premiere.
Expect to see an increasing number of sites using video as a standard background in future. The new sample theme for WordPress, Twenty Seventeen, has a video header which is likely to inspire a lot of designers to go down this path.