We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Émery [3], the exponential curvature-dimension condition of Bauer et al.[5] and the coarse Ricci curvature of Ollivier [34]. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an L 1 transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Meyer's theorem under the curvature-dimension condition CD(κ, ∞) of Bakry and Émery [3].