Abstract We describe, test, and apply a technique to incorporate full-Sun, surface flux evolution into an MHD model of the global solar corona. Requiring only maps of the evolving surface flux, our method is similar to that of Lionello et al., but we introduce two ways to correct the electric field at the lower boundary to mitigate spurious currents. We verify the accuracy of our procedures by comparing to a reference simulation, driven with known flows and electric fields. We then present a thermodynamic MHD calculation lasting one solar rotation driven by maps from the magnetic flux evolution model of Schrijver & DeRosa. The dynamic, time-dependent nature of the model corona is illustrated by examining the evolution of the open flux boundaries and forward-modeled EUV emission, which evolve in response to surface flows and the emergence and cancellation flux. Although our main goal is to present the method, we briefly investigate the relevance of this evolution to properties of the slow solar wind, examining the mapping of dipped field lines to the topological signatures of the “S-Web” and comparing charge state ratios computed in the time-dependently driven run to a steady-state equivalent. Interestingly, we find that driving on its own does not significantly improve the charge state ratios, at least in this modest resolution run that injects minimal helicity. Still, many aspects of the time-dependently driven model cannot be captured with traditional steady-state methods, and such a technique may be particularly relevant for the next generation of solar wind and coronal mass ejection models.