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The magnetorotational instability (MRI) is an important process in sufficiently ionized accretion disks, as it can create turbulence that acts as an effective viscosity, mediating angular momentum transport. Due to its local nature, it is often analyzed in the shearing box approximation with Eulerian methods, which otherwise would suffer from large advection errors in global disk simulations. In this work, we report on an extensive study that applies the quasi-Lagrangian, moving-mesh code AREPO, combined with the Dedner cleaning scheme to control deviations from $\nabla\cdot B=0$, to the problem of magnetized flows in shearing boxes. We find that we can resolve the analytical linear growth rate of the MRI with mean background magnetic field well. In the zero net flux case, there is a threshold value for the strength of the divergence cleaning above which the turbulence eventually dies out, and in contrast to previous Eulerian simulations, the strength of the MRI does not decrease with increasing resolution. In boxes with larger vertical aspect ratio we find a mean-field dynamo, as well as an active shear current effect that can sustain MRI turbulence for at least 200 orbits. In stratified simulations, we obtain an active $αω$ dynamo and the characteristic butterfly diagram. Our results compare well with previous results obtained with static grid codes such as ATHENA. We thus conclude that AREPO represents an attractive approach for global disk simulations due to its quasi-Lagrangian nature, and for shearing box simulations with large density variations due to its continuously adaptive resolution.