What's wrong with the hydraulic analogy? When one understands atomic theory, one knows that liquid water is composed of *illions of molecules banging around at about 1500 m/s. Almost any flow rate you'll encounter, is a small fraction of this, and so we can make the same observation that pressure is transmitted from molecules bumping into each other, not so much the overall flow; and that the dominant motion is internal and random.
It's an uninteresting argument, and worst of all, doesn't illuminate anything to the reader. So what if it's mostly random motion, does that just mean it's getting hot or something? No, that's just thermal energy. Oh, and it's too small to see. Well so what. (Well, yes actually, it is effectively the origin of resistance. But proving it gives a proportional [Ohm's] law, is much harder. Or understanding the sequence of facts that are required to derive such a law -- that's whole-ass statistical mechanics, arguably.
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What's interesting is when the charges are "free" to flow, i.e. in free space under the influence of fields, ballistically, rather than damped out by random motions. Then we find a V^(3/2) power (Child-Langmuir) law, we find mass and we find everything that we expect as a consequence from that (i.e. all the electron-resonant microwave tubes).
Cool fact, the electrode (especially grid) capacitance of a late model vacuum tube about doubles going from cold, to hot and biased. The cathode electron cloud expands and contracts in response to the grid's electric field, giving and taking energy from the electric field; mass is a conservative property, and so too is the manifestation at the electrode -- reactance. If some of the cloud happens to leave, however, its initial energy is carried off with it -- beam current causes loss, manifest as grid conductance/resistance, however you like to express it. This extends to subsequent electrodes, too: the screen, suppressor, etc. experience a small transconductance, from grid to grid, even when the later grid is reverse-biased (as suppressor or pentagrid control 2). The reciprocal is not true, however: the beam acts as a conveyor belt, breaking symmetry, and thus this is a nonreciprocal property (gm(1-3) nonzero but gm(3-1) zero, or, effectively just plain old interelectrode capacitance because it's not going to be |zero| in practice).
This property is even used to advantage in some types: dual-control pentodes (6HZ6 etc.) and sheet pentodes (6BN6 etc.) can be used for FM discriminator service, where g3 is simply tied to GND through a high impedance parallel LC tank. gm13 is quite small (10s umho?) so the impedance needs to be quite high (100s kohm), but given this, it resonates sympathetically with the driven source. g3 voltage acts to gate the output (plate) current, so the plate current ends up PWM with respect to the phase shift between source and resonant tank. And that phase shift is proportional to frequency shift vs. resonance (for small enough changes in frequency -- for commercial WBFM, the Q needs to be fairly low; like I said, high impedance tank!).
Ballistic transport is also possible in some semiconductors; in others, avalanche breakdown occurs first, but when it doesn't, current flattens out, or even goes negative (as is the case for GaAs (Gunn diodes), something about varying saturation velocity in different Brillouin zones, i.e. the electrons changing directions as they travel through the crystal).
Tim