When a capacitor is connected to a DC circuit, what ensures that the current on both sides of the capacitor is the same? When charges arrive at one end of the capacitor they stop moving; presumably they give their kinetic energy to charges on the other side of the capacitor so they can leave. How exactly does this transfer of energy occur?
You seem to be asking two questions: why is the current on either side the same, and what happens to the energy. To clarify where the energy goes, the kinetic energy is transferred entirely into electrical potential energy of the capacitor. This is the energy stored as a result of all the similar charges being close together on either capacitor plate. There is no actual energy transfer from one plate to the other.
For the currents, let's imagine the current on one plate was larger than the other. The capacitor would end up building a net charge, as a result of the charge on one plate being larger than the charge on the other. By conservation of charge, this extra charge must have come from somewhere else in the circuit, and there must be the opposite charge to cancel out the net charge on the capacitor. This is an energetically unfavourable situation, so doesn't happen without some additional influence. We can thus conclude that the current on either side is identical.