I grow many varieties of mulberries. The cuttings will push fruit along with the leaves, however they will not ripen and it's better if you clip them off. They're more difficult to root than figs. Out of 10 cuttings you may end up with 2 or 3. The best method for propagating them is through grafting.
Here's two varieties I grafted on same tree.
The white one is called Greece, The black one is Collier.