Effect of electron discrete nature on current density
Current through a cross section is

$$I = \int_A \left(\sigma \vec{E} + \varepsilon \frac{\partial \vec{E}}{\partial t}\right) \cdot \mathrm{d}\vec{A}$$

The first term in the sum is for conductors, their conductance sigma is that high the second term does not matter. You can't "saturate" conductors electrically. Before that happens, you saturate them magnetically. (But that's another story.)

The second term in the sum is for isolators. Their sigma is so tiny it does matter. It's the dielectric displacement current. You can see the dielectric constant epsilon and the change of the electric field over time rather than the electric field matters.

If you have boundaries, the electron work function on the conductor also matters. This is e.g. important for tubes, but also for semiconductors.