  We give an alternative proof for discrete Brunn-Minkowski type inequalities,
recently obtained by Halikias, Klartag and the author. This proof also implies
somewhat stronger weighted versions of these inequalities. Our approach
generalizes ideas of Gozlan, Roberto, Samson and Tetali from the theory of
measure transportation and provides new displacement convexity of entropy type
inequalities for the lattice point enumerator.
