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Abstract: In this paper we present an extension of the Dynamic-Monge-Kantorovich model that considers a time derivative of the transport density that grows as a power law of the transport flux counterbalanced by a linear decay term that maintains the density bounded. A sub-linear growth penalizes the flux intensity (i.e. the transport density) and promotes distributed transport. Corresponding equilibrium solutions are reminiscent of Congested Transport Problems. In this case, we are able to relate our proposed dynamics with $p$-Poisson equations typical of congested transport. On the contrary, a super-linear growth favors flux intensity and promotes concentrated transport, leading to the emergence of steady-state "singular" and "fractal-like" configurations that resemble those of the Branched Transport Problem. In this case we are able to offer a Lyapunov candidate fucntional that is minimized at equilibrium. Using the necessary caution, we conjecture that the proposed extended dynamic Monge-Kantorovich problem for $β>1$ is indeed a version of a branched transport model. In all situations, we derive a numerical discretization of our model that is accurate, efficient, and robust for a wide range of scenarios. Using ad-hoc linear solver strategies, we are able to obtain numerical approximations that show highly irregular and fractal-like formations.