In this talk, we will demonstrate how the celebrated connection between Ricci curvature, optimal transport, and geometric inequalities such as the Brunn-Minkowski inequality, extends to the setting of general Lagrangians on weighted manifolds. As applications, we will state a generalization of the horocyclic Brunn-Minkowski inequality to complex hyperbolic space of arbitrary dimension, and a new Brunn-Minkowski inequality for contact magnetic geodesics on odd-dimensional spheres. The main technical tool is a generalization of Klartag's needle decomposition technique to the Lagrangian setting. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027)