Here is a 24 minute long tutorial of how to do it
There are also android and apple apps, that are a lot cheaper, and make it easy. They aren't always as good, but can't beat the ease of them.
This is called a cinemagraph. Though this particular type uses a motion vector "optical flow" approach with specialized software. Check out Plotagraph Pro. It's kind of related to morphing.
Here's a good example: http://trey-ratcliff.squarespace.com/
If you're wanting a more traditional, simple loop, you can try Flixel.
Flixel.com automates the process, or you can do it manually. Phlearn on Youtube has a tutorial on it.
Wallpaper was from here and is this one. Unfortunately not in 4k like the tag suggests, but still looks good nonetheless.
Skins are from here.
I was recently looking into Cinemagraphs again and it's amazing how some are just really high quality and would look nice on a big screen looping for a few minutes. Since hooking into Daydream is probably not possible it would still be nice to have an app to manually run. It would then start to show the best (currated?) Cinemagraphs on TV. Slowly fading to black and to the next one every few minutes. It could also be a phone app that uses Chromecast functionality to start the slideshow.
Examples: https://flixel.com/cinemagraphs/fresh/
I downloaded this cinemagraph to use as a background with this widget, then overlaid this moon-phase widget (make sure to set the iconSet variable to "shadow").
There are programs devoted to doing just that, https://flixel.com for example. Never used them though. It's usually just masking the still shot as a layer over the animated part in Photoshop or After Effects.
I bought it at a photo show, the discount brought it down to £50 (that's US$76 today). I bought the pro mac version (the ipad version, which I got originally, isn't nearly as useful: too many compromises). I used the Mac version on some timelapse movies I shot: https://flixel.com/cinemagraph/pxsg7sqt8er4w4i279xh/ and https://flixel.com/cinemagraph/ejr4xd4jojswd4538g47/ Decent software, not entirely sold on their hosting charges though.
You won't find the actual cinemagraphs on the site, just the images/videos from which they are derived.
Try here instead: https://flixel.com/Travel_Alberta/ https://www.youtube.com/user/TravelAlbertaCanada http://travelalberta.tumblr.com/