It took me many years to figure out and develop ways to teach my students how to play extensions. I now have a system of starting with a Bb major scale using tunnels and swinging the left elbow forward and back depending on the finger being played. This way of learning is for playing without vibrato and is also used when playing fast. For slower tempos that use vibrato, the thumb will move much more as the left hand changes finger.
What I came to realize is that depending on the speed of the music, you need to think about these aspects:
I teach the "Backward" extension first which you can see in the video below and then usually wait at least a few weeks before teaching the "Forward" Extension. This gives time to get used to using the "side of the tip" of the first finger which is such a new feeling compared to playing first finger when it is not in extension.
I have learned that it takes a long time for extensions to settle in and become automatic in a student's playing. Students will forget to move their thumb in forward extensions. You'll need to keep reminding them and trying backward and forward extensions in different pieces and etudes for probably a year or more! There are some wonderful extension etudes in 25 Etudes by Rudolf Matz. I especially love #17.
The first part of the video below is the end of a lesson when the parent was recording what I had just taught a student - how to do a backward extension. From 1:00 to 1:45, I show a great trick from Cornelia Watkins which gives the student the feel of the tip of their first finger. This is so important for them to discover!