We demonstrate that the direction and values of the radiative heat flux exchanged between a non-phase-change material and a phase-change one excited with a temperature difference modulated in time can efficiently be tuned by means of their common steady-state temperature. This heat-flux modulation occurs in both the far- and near-field regimes as a result of the strong temperature dependence of the emissivity and permittivity of the phase-change material, respectively. It is shown that the heat pumping into or out of the phase-change material can not only be amplified but also canceled out for temperatures around its critical temperature. This nullification of the radiative heat flux can be used as a mechanism to rectify heat currents and to insulate the two bodies from each other, even when their temporal temperature difference is different than zero. The obtained results thus open a new pathway for the heat-flux control of nonequilibrium radiating systems.