Actually, when you have electric current, you must see it as a flow of electron. It does not work in the way you explain. Yes, free electrons do re-combine with the atoms core (the 'holes'), but when a voltage is applied across a conductor, electrons also get 'ripped' from the atoms and become loose and flow to the next atoms.
So, let's say you have a chain of atoms in a conductor, and you put a voltage across the conductor, then, the positive side will attract atoms from the negative side. Atoms will start flowing from the negative side, and hop to the next atoms. Electrons from those atoms will hop to the next, and the next, and to the positive side.
Also, even though the current goes to very high speed (near the spead of light), the electrons does not flow anywhere near that speed. How come? Well, let's see an analogy with an ordinary garden hose. Let's suppose you have your hose which is ALREADY FILLED with water (see it as electrons already in a wire). Then, on one side, you put some pressure (you start a pump for example). Current will flow through the hose (water current). However, at the moment you turn the pump on one side, this create a pressure wave that travel very fast through the hose. You can see almost immediately water coming out at the other end. However, the water you first see comming at the other end is the the water entering the hose when you started the pump. Water can flow really slowly in the hose. The same is applicable for current.
One law of electricity is that the sum of the current entering a device is equal to the sum of the current leaving a device. So, the current entering the emitter is the same as the sum of the current leaving the base or collector.