A remark on discrete Brunn–Minkowski type inequalities via transportation of measure

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 on the n-dimensional integer lattice.