Only the part within the yellow square 'flexes' up an down.
Nothing 'moves' in the sense that a piston in a cylinder moves kind of way.
At least it shouldn't....

Don't know if the membrane needs to be 'burnt in'.
In the 3 ones I had from new I never noticed much difference but others may have opposite experiences.
The ones I have (had) never stayed 'stock' for long. 
Never modified the driver though only 'motivate' it to do things in a different way and damp it externally which worked for me.