We introduce curvature-dimension conditions for metric measures spaces where the lower curvature bound is a lower semi-continuous function. The curvature-dimension conditions admit geometric and analytic consequences as a generalized Bonnet-Myers theorem or refined contraction estimates for entropy gradient flow curves. Moreover, there is stability w.r.t. (pointed) measured Gromov-(Hausdorff) convergence and suitable convergence of the curvature functions.