Reddit reviews The Invisible Art: The Legends of Movie Matte Painting

The Invisible Art is the most famous piece of illustrated content about matte painting (so... set extension)
There is not a lot to say about methodology... production designers build a set in accordance with the director, scripts, concept artists and budget. The scope of the scene might require bigger sets than the production designer can build under budget. So the build is kept to a minimum and the VFX team comes in to "extend" the set to match the directors vision. Back in the days this required a locked camera and a matte painter would come in to paint - possibly on glass- the added elements, but nowadays it happens completely in post, digitally.
The camera movement from the plate is analysed by match movers, a LIDAR scan might be used to reproduce a high density mesh of the scene, a 3D version of the scene is inferred from this data. The digital matte painters (names vary depending on the company they are in) model extra geometries, paint details, patch trees in the background... whatever is necessary to build the end shot. This is then rendered one way or another through the shot camera and compositors work to smoothly blend the real plate set with the CGI one, using all the tricks in their bags. Not sure if this answers your questions.