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kwired's description is good. Just consider that what is described as current flow is really a flow of energy.
If you go inside a vacuum tube with an electron emitter (cathode) and a plate electrode (an electron collector when positive relative to the cathode), then there is an actual flow of electrons from the cathode to anode. The velocity of this flow is a function of the electron charge, its mass, and the electric accelerating field (plate to cathode voltage). So the electrons do not move instantaneously from the cathode to the plate.
Back to the wire. There is a gradual average flow of electrons from one end to the other, but this is quite slow in comparison to the flow rate of energy. If you introduce a pulse of energy into one end of a wire pair the pulse will travel at about 0.6 to 0.8 times the speed of light. The velocity of light in free space is about 186,000 miles per second, or 982 feet per microsecond.
My photo P3 at
http://beta-a2.com/cat-5e_photo.html shows 1.5 microseconds for the propagation time on a 1000 ft length of CAT-5E cable. This is a velocity of 667 ft/microsecond or about 0.68 times the speed of light.
Dow discusses vacuum tube electron transit time in chapter 2 of "Fundamentals of Engineering Electronics", 2nd Edition. Tube dimensions became important in the VHF frequency range because of this transit time.
For normal circuit analysis at low frequencies it is convenient to speak of current flow as being instantaneous, and it really does not matter whether or not that a single individual electron instantly goes around the path.
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