capacitor with varyable permittivity dielectric

Ok so I have a question , I have thought about it and I have heard some answers from other poeple so I would gladly hear some from the members of this forums.

Say we have a DC source and a transformer primary in series with that source , and then we add a capacitor in series with the transformer pirmary.Normally when you connect the dc source what happens is at first there is a current flow due to the charging up of the capacitor , once the capacitor is charged the current flow stops and the system reaches equilibrium, since a transformer can only transfer energy as long as the current in the primary is either changing amplitude and or polarity then after the cap charges up the transformer cannot induce a current in the secondary anymore.

Now what would happen if one could have a dielectric material between the plates such that one could alter it's dielectric constant while the cap is charged , so with varying the constant one would also change the capacitance of the capacitor , and with either lovering or increasing the capacitance and keeping voltage the same would result in charge flow back and forth from the capacitor and such charge flow could make the transformer do useful work even with a dc supply , am I correct or is there something wrong here ?