We will study metric measure spaces (X, d, m) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution- valued lower Ricci bounds BE_1(\kappa, \infty)

* for which we prove the equivalence with sharp gradient estimates,

* the class of which will be preserved under time changes with arbitrary \psi \in Lip_b(X), and

* which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets Y \subset X.

In the latter case, the distribution-valued Ricci bound will be given by the signed measure \kappa = k m_Y + l \sigma_{\partial Y} where k denotes a variable synthetic lower bound for the Ricci curvature of X and l denotes a lower bound for the "curvature of the boundary" of Y, defined in purely metric terms.