For me, the main thing that hasn't been mentioned is this: the force that stops the strings doesn't come from your hand or arm muscles, it's the weight of your arm. Your left thumb especially shouldn't be doing a lot of work. Really, all your muscles have to do is hold your arm in shape (which your posture muscles are perfectly capable of doing, even under that much load), and 'walk' your arm up and down the fingerboard (which your twitch muscles can do easily)... leave stopping the strings up to gravity (your arm weighs about twice the stop pressure, and if you start adding your torso to that you can easily develop the power required).
The other great thing about using gravity like this is that when you start doing transition and thumb positions, it's still completely natural... the weight of your arm is still falling through your fingers or thumb, it's just on a different angle.