This work is concerned with Damage Transport (DT), a framework for continuum damage mechanics that provides softening damage models while ensuring mathematically well-posed boundary value problems. DT was constructed in a two-step process: First, the cross-convex envelope of the time-incremental potential was calculated analytically for a simple rate-limited scalar damage model. In a second step, an entropic regularization term was added to the dissipation to ensure uniqueness of solutions to the incremental problems. A key characteristic of DT is the natural emergence of damage distributions as internal variables, a consequence of the cross-convex relaxation. In particular, the space of internal variables of the model is inherently infinite-dimensional. If the evolution of the material starts from an undamaged state, it was shown that an implicit Euler discretization of DT in time leads to solutions which are finite sums of Dirac measures. Thus, only finitely many degrees of freedom are needed. However, the attained damage states depend on the time step size and on the number of time steps. Even more critically, previous constructions were based on a regular time discretization, and the number of time steps was required to be known a priori. In particular, an adaptive discretization in time, not uncommon for industrial finite element codes, is incompatible with the model developed so far. This work proposes a strategy to overcome this limitation: For a fixed number of attainable damage states, the damage values and the damage fractions are selected in an adaptive manner from one time step to the next. In particular, the number of internal variables is fixed a priori and adaptive time stepping is enabled. The proposed strategy is based on a simple postprocessing step, where newly created damage states are redistributed to neighboring damage states. This redistribution follows a greedy strategy and is selected to exactly preserve the (damaged) elastic stiffness. As the contribution is based on postprocessing, the efficient computational framework previously developed remains applicable. Also, we show that the redistributed damage transport model satisfies the Clausius-Duhem inequality, i.e., the model is thermomechanically consistent. We demonstrate the accuracy and versatility of the novel framework via computational examples.