CBSE Class 12 · 2012 · EM WavesShort Answer

There is no conducting medium between the plates, so no conduction current can cross the gap.
With a dc source, once the capacitor is charged the field is steady, $d{\Phi }_E/dt=0$, so nothing flows.
With an ac source the charge on the plates, and hence the electric field and flux ${\Phi }_E$ between them, change continuously. Maxwell showed this changing flux is equivalent to a current --- the displacement current

which equals the conduction current in the wires, so current appears to flow continuously.