How should I do this?

I’m trying to rig a cartoon mouth using shape keys, but I’m having trouble making the mouth move and having terrible distortion as seen in this file (right). Is there a way to use proportional editing (while having multiple vertices selected) to make the mouth frown, while keeping the mouth on the general curve of the mesh? I know for single vertices you can use the g + g keyboard command, but I have a whole area I want to move. Any tips are welcome!

first select the vertices, before you press g,
press o to turn on the proportional editing (or the circle at top- middle of the screen, turn it on) (2.8)

you will then see a circle of effect, sometimes its too big (out of the screen)
press page-up and down to change the size of the circle, change it to moderate size
and then press g to move the vertices, the circle is the area of effect,
you can page-up and down while moving real time.

also open the circle button menu mentioned before, if you choose “connected” mode,
this will only move vertices that are linked first.
useful for mouth and lips for example, where vertices are very close together
and you only want to move the top lip but not the bottom.

Well, I am well acquainted with proportional editing, but I’d like to keep the movement to a certain range, as if I were moving a single verticy on a sphere by pressing g twice. The failed example in my file actually did use proportional editing.