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In this paper, we present the extension of the space-time expansion discontinuous Galerkin to handle ideal and viscous magnetohydrodynamics (MHD) equations. The local time-stepping strategy that this scheme is capable of allows each cell to have its own time step whereas the high order of accuracy in time is retained. This may significantly speed up calculations. The diffusive flux is evaluated through a so-called diffusive generalized Riemann problem. The divergence constraint of the MHD equations is addressed, and a hyperbolic cleaning method is shown that can be enhanced by utilizing the local time-stepping framework. MHD problems such as the Orszag-Tang vortex or the magnetic blast problem are performed to challenge the capabilities of the proposed space-time expansion scheme.