The confusion here is from the initial poor description of how a battery works.
A battery consists of three things: a positive electrode, a negative electrode, and an electrolyte in between. The electrodes are made of materials that strongly want to react with each other; they are kept apart by the electrolyte.
The electrolyte acts like a filter that blocks the flow of electrons, but allows ions (positively charged atoms from the electrodes) to pass through. If the battery is not connected to anything, the chemical force is pulling on the ions, trying to draw them across the electrolyte to complete the reaction, but this is balanced by the electrostatic force-- the voltage between the electrodes. Remember-- a voltage between two points means there is an electric field between those points which pushes charged particles in one direction.
When you add a wire between the ends of the batteries, electrons can pass through the wire, driven by the voltage. This reduces the electrostatic force, so ions can pass through the electrolyte. As the battery is discharged, ions move from one electrode to the other, and the chemical reaction proceeds until one of the electrodes is used up.
Thinking about two batteries next to each other, linked by one wire-- there is no voltage between the two batteries, so there is no force to drive electrons. In each battery, the electrostatic force balances the chemical force, and the battery stays at steady state.
(I kind of glossed over what it means for two materials to "want" to react with each other. Google "Gibbs free energy" for more details on that. You might also google "Nernst equation.")
Individual electrons move around in semi-random ways, with a bias of moving along the E field lines from negative to positive. "Current" is only predictable because many many electrons are involved, and in the aggregate, their random behavior largely averages out and we are left with just the E field effects on the herd.
From this view it should be clear that individual electrons can end up lots of different places after a certain amount of charge goes by. For a sustained current such as you describe, many electrons are moving all along the closed conduction path. Some may indeed swap positions after some amount of charge has flowed.
None of this has anything to do with semiconductors. The same thing applies to electrons in a resistor or a copper wire.
Best Answer
Batteries store chemical energy. The two different metals and the electrolyte react in such a way that electrons will flow through an external circuit from the negative terminal of the battery to the positive terminal. As the reaction proceeds, the metals are converted to another material and the electrolyte becomes weaker until finally, the reaction stops.
For example, in a charged lead acid cell, we have:
The electrolyte reacts with the lead plate to convert some of it to lead sulfate along with some excess electrons.
The electrolyte reacts with the lead oxide plate to convert some of it to lead sulfate along with a deficit of electrons.
With an open circuit, the reactions are stopped due to the charge accumulated on the plates.
When an external circuit is present, the excess electrons from the lead plate flow through the external circuit to the lead oxide plate but this allows the reactions to proceed anew and the process continues until the cell is discharged:
So, the battery drains because chemical reactions drive the electrons through the external circuit until the reactants are exhausted.
(images from Wikipedia article for "Lead-acid battery")