Thanks. Didn't know you call it "to shift weight"
As for the technique, I think there are many ways of doing punches and the way I teach wasn't invented by myself ) I was taught it long ago in the Soviet time.
Of course, we don't shift weight on the front foot like they do in karate. Maybe it's better to say we lean forward a bit when punching in order to reach longer and when the jab is completed we bend our arm in the previous position and put down the right heel at the same time and it helps us to avoid the opponent's attack. This is what we call "to shift weight" in Russian

by the way, what do you say in English - punch lighter/harder or stronger/softer

I'm just not familiar with this terminology )