We consider the functional of total variation of maps valued in a Riemannian manifold (MTV). The L2-gradient flow of MTV was proposed as a regularizing method for manifold-constrained data, such as the color component of images. However, well-posedness of the flow for generic manifolds is out of scope of general theory due to lack of geodesic (semi)convexity of MTV. An alternative approach, used before for p-harmonic map flows, p>1, is to apply nonlinear parabolic PDE theory to the PDE system formally describing the flow. This is non-trivial, as the system is strongly nonlinear, singular and degenerate. I will report on recent results on existence, uniqueness and large-time behavior of solutions to the system obtained in collaboration with L. Giacomelli and S. Moll.