  We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric
diffusion operators by convexity of entropy on the $L^2$-Wasserstein space, and
define a curvature-dimension condition for general metric measure spaces
together with a square integrable $1$-form in the sense of
\cite{giglinonsmooth}. This extends the Lott-Sturm-Villani approach for lower
Ricci curvature bounds of metric measure spaces. In generalized smooth context,
consequences are new Bishop-Gromov estimates, pre-compactness under measured
Gromov-Hausdorff convergence, and a Bonnet-Myers theorem that generalizes
previous results by Kuwada \cite{kuwadamaximaldiameter}. We show that
$N$-warped products together with lifted vector fields satisfy the
curvature-dimension condition. For smooth Riemannian manifolds we derive an
evolution variational inequality and contraction estimates for the dual
semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada
\cite{kuwadaduality, kuwadaspacetime} yields Bakry-Emery gradient estimates.
