So what you need to do is to cut the branch just slightly under the new shoots.
(Sorry, the pic is blurry!)
And just poke the newly cut branch into any soil and 99%, the new shoot will survive and you will have mulberries growing pretty soon.
That's the very simple way to propagate mulberry trees. I have given away a few pots of mulberry trees and also been collecting new babies so I can bring them over to my new place soon.^^
p/s: I also used this same method to propagate my Hydrangeas. Worked all the time. :D