A variational damage model is presented in which the macroscopic material response emerges from the evolution of homogenized damaged microstructures subject to explicit rate constraints. The formulation is based on an incremental energy minimization principle, where the reduced energy density consists of a rank-one convexified free energy density and a dissipation term defined via the Wasserstein--1 distance between distributions of internal damage variables. To control admissible microstructural evolution, a rate constraint is imposed on the dissipation potential [1]. Damage growth is allowed only within a prescribed rate window, while all other rates result in infinite dissipation and are therefore excluded from the variational principle. This modification induces a rate-restricted dissipation distance and, consequently, a constrained (semi-)convexified incremental condensed energy. Thus, (semi-)convexification procedures are presented that respect the dissipation constraint in the high dimensional space of the deformation gradient. The proposed framework establishes a rigorous link between constrained microscopic evolution and macroscopic rate effects. It naturally fits into relaxation-based approaches to damage and admits a clear interpretation in terms of optimal transport [2] of internal variables. The model is well suited for numerical implementation within variational finite element formulations and provides a flexible basis for the simulation of rate-controlled damage processes in computational solid mechanics.