Uniform Ricci curvature bounds are a now classical tool for proving convergence to equilibrium for Markov processes, as well as functional and concentration inequalities in presence of convexity. In this talk, I will discuss some results obtained when using non-uniform lower bounds. Applications include a self-improvement of the Bakry-Emery estimate of Poincar\'e constants of uniformly log-concave measures, and convergence to equilibrium in Wasserstein distance for diffusion processes. Joint work with P. Cattiaux and A. Guillin.