The animated models in that thread come straight from Pokemon Showdown, which has a collection of animated and static images for both Shiny and non-Shiny, front and back-facing Pokemon. They can be found in the public /sprites directory as seen here. If you need front-facing, standard animated models, check /xyani.
The URL https://play.pokemonshowdown.com/sprites/xyani/INSERTNAME.gif works for every Pokemon if you add the Pokemon's name into it correctly, which is a quick way to get the links on the fly. (Despite the name of the /xyani path, Gen 7 models can also be acquired by doing this.)
Back when X and Y first released and the 3D models were all new, Spanish fansite pkparaiso.com was the go-to for getting animated pics, and this remains the case if you're interested in the battle animations for each Pokemon. The site isn't updated for US/UM, but you can still get some cool stuff:
