  We study various transport-information inequalities under three different
notions of Ricci curvature in the discrete setting: the curvature-dimension
condition of Bakry and \'Emery, the exponential curvature-dimension condition
of Bauer \textit{et al.} and the coarse Ricci curvature of Ollivier. 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$(\kappa,\infty)$ of Bakry and \'Emery.
