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We study the dynamic evolution of the charge distribution (polarization) of a 2×2 quantum-dot cell with two electrons in the presence of a time-dependent driver cell and a magnetic field. We describe the effects of the magnetic flux on the response of the basic dot cell, for fixed, and linear switching of the driver polarization. In the static case, we find that the magnetic field has a strong localizing effect, similar to the effect of asymmetry. For fixed tunneling, the polarization of the target cell increases with magnetic field, going through a maximum at a particular value of the magnetic flux through the cell. In the dynamic case, a ringing effect and a decrease in the final polarization value of the target cell are obtained as the magnetic field increases. The effects of temperature and asymmetry on these results are also analyzed.