Thank you!
Your question: I don't know. That's what we are ultimately trying to determine, but at this point all we know is that there is an energy transfer. What form that energy takes is something to be figured out.
Hopefully you now agree there is energy transfer across that gap. If not, how can one side insert some and the other side use some? We are still not talking about how it transfers, only that there is a transfer.
You are trying to determine that as for me is very clear. The energy is not transferred across the gap it is transferred in to capacitor so charge particles accumulate on one wire / plate and equal amount of charges leave the other wire / plate.
Hope you agree that no electrons jump that gap. Since current is defined as flow of electric charged particles and since there are no electrons or ions traveling through that gap there is no energy transfer through that gap.
In order to charge a capacitor you will need to move charge particles (electrons) to and from plates (you move electrons to one plate and simultaneously remove electrons from the other). The electrons you remove from that other plate are there when capacitor is charged and do not come from the other side.
This was the reason I chose to use the two parallel capacitors example as there you have nothing else just capacitors one it is charged and the other is not so you have two gaps in the loop and no electrons travel across any of the two gaps yet there is an electric current in the wire so electrons move from the plate of one capacitor to the plate of the other capacitor through the wire when switch is closed.
On the charged capacitor there is excess of electrons on one plate and deficit of electrons on the other plate.
When you connect the other identical but discharged capacitor in parallel electrons will flow through the wire from one capacitor to the other.
So from the plate with excess electrons on the charged capacitor the electrons will travel to the plate with neutral charge on the discharged capacitor and at the same time electrons from the neutral charge on the discharged capacitors will move to the plate with deficit of electrons on the charged capacitor.
So if you were not to look at what happens to electrons in the capacitor plates you will think that those electrons must have jumped the gap but that is not the case.
In a circuit with resistance energy will be lost when part of the charge is moved from one capacitor to another.