Displacement current between grid and cathode on heated electron tubes?

See previous post on supplying a Nuvistor electron tube with heater current.


The the cathode starts emitting electrons, the grid becomes charged negative.


The difference between a conventional capacitor in series with a resistor and a voltage source and this case is that here we have the voltage source to be the capacitor itself as the heated cathode emits electrons.


Is there displacement current?


On an ordinary capacitor there is not (internal current between the plates). But it behaves as if there is.  Between the plates there is a magnetic field created by the existence of a varying electric field. See Maxwell electromagnetic theory.


How about the electron tube? Are electrons internaly transported from cathode to grid. Or is it an outside leakage current?