If you put a neon light tube inside an electric field such that it is aligned with the field lines, it will start to emit light. This is caused by a current starting to flow inside the tube.
This current are moving electrons. But I am wondering, should there not be a point when all free moving electrons have moved to one side? And when this "equilibrium" is reached, no light should be emitted anymore. But that is obviously not the case? But why?
This is demonstrated in these lectures.
The current is indeed moving electrons.
The primary answer to your question is: the electrons involved in the current are eventually moving into the ground. The intensity of the current depends on the impedance offered by the path to the ground. This is why the intensity of the light is low when the path the current is taking is through the insulated holder while it is higher when it is going through the somewhat less insulated gloves+person holding the lamp.
If the source of the static charge was simply a fixed charged metal piece, you'd be right. The light intensity would exponentially reduce (similar to a capacitor discharge).
However, in this case, the source appears to be a Van De Graff generator. As soon as the static electricity is removed from the surface, some mechanism replenishes it. In an old Van De Graff, it would've been some belt based mechanism. We can't be sure what it was in the video you have in your post.
