The backgrounds are a flat plane that extends into the background, and is then replaced by a prerendered scrolling 'matte painting' background plane. The background in 'Monument Valley' is animated, it has a setting sun.
The 'Acropolis' stage has two background layers, creating a subtle parallax effect.
Screenshots marked with 🍒 are created by syltefar and are considered public domain, free to use for anything. If you want to, you can note where you found it and link to this page.