I think the OP might be asking how does a potential difference exist at the gate of the FET since there is no apparent place for current to flow. With the caveat that I haven't been anywhere near a course on this stuff in 30+ years, [nevermind, I just looked into FETs more closely, and my understanding was incorrect]

With a bipolar transistor (I'm on more familiar ground here), s small amount of current applied to the base-collector pair of pins allows a much larger amount of current to flow from the emitter to the collector (at least for NPN transistors). At least that's what I remember from the rudimentary electronics I learned in high school.

I now know that the gate, source and drain of a FET are roughly equivalent to the base, emitter, and collector of a BJT (thanks wikipedia).

Actually, nothing "moves" inside a transistor in the practical sense. At the atomic level, charge is being transferred, but nothing is moving. Magnetic movement based on current would be a relay (which was used for switching before transistors).

Voltage isn't being applied to a point. All transistors have some current flow from the gate (or base) to either the drain (collector), the source (emitter), or if available the body (substrate). The actual configuration of the current flow is based on the circuit, application, and particular type of transistor used.