There must be an electric field applied, and a path for charge to flow, in order to move charge from one plate to the other.
As an analogy, picture a plastic cube half full of water and open on top. There is barrier down the middle, so that each side is separated from the other. There is a plastic tube that comes out of the bottom of each side, and goes to each side of a bi-directional pump. The pump can move water from one side to the other. If the water level is the same, there isn't any energy in the system. If the pump moves water from one side to the other, there is now stored energy (the force from gravity that creates pressure on one side) that can move the water back to the other side if the pump shuts off (assuming the pump is "leaky" when turned off). Water moves from one side to the other, but doesn't go through the barrier in the middle.
This is "sort of" what happens with a capacitor - though obviously the physics are different since we are talking about electromagnetic fields instead of gravity. Instead of water it is charged particles, and instead of tubes there are wires, and instead of a pump there is a voltage source. At first the charged particles are evenly distributed on both plates of the capacitor. The voltage source provides an electric field that moves the charged particles from one plate to the other. One plate becomes more positive, and the other, more negative. The difference in charge on the two plates causes an electric field to form between them. The charged particles are moved until the electric field across the barrier between the two plates is equal and opposite that which is being applied by the voltage source. Now if the voltage source turns off (but again, is "leaky" when off), the electric field across the plates of the capacitor is such that it provides a force on the charged particles that will move them back to the other side, until the charge on the two plates is balanced again.