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In this note we continue the analysis of metric measure space with variablericci curvature bounds. First, we study $(\kappa,N)$-convex functions on metricspaces where $\kappa$ is a lower semi-continuous function, and gradient flowcurves in the sense of a new evolution variational inequality that captures theinformation that is provided by $\kappa$. Then, in the spirit of previous workby Erbar, Kuwada and Sturm \cite{erbarkuwadasturm} we introduce an entropiccurvature-dimension condition $CD^e(\kappa,N)$ for metric measure spaces andlower semi-continuous $\kappa$. This condition is stable with respect to Gromovconvergence and we show that is equivalent to the reduced curvature-dimensioncondition $CD^*(\kappa,N)$ provided the space is non-branching. Finally, weintroduce a Riemannian curvature-dimension condition in terms of an evolutionvariational inequality on the Wasserstein space. A consequence is a newdifferential Wasserstein contraction estimate.

Evolution variational inequality and Wasserstein control in variable curvature context