First, this question is different than Does an electrolytic capacitor have a minimum voltage? since that question is about circuit analysis.
So, my question is the following:
Let's pick two voltages, a high one and a low one. Let's say 5kV and 30V.
Let's imagine a capacitor made for a circuit operating at 5kV; if the dielectric is too thin, the charge may jump across; so the capacitor is designed to keep charges with a 5kV potential apart.
But if the dielectric is thick enough to handle higher voltages, how can a lower voltage, like 30V, attract a charge on the other side?