What is the current in a FET as gate voltage is increased towards large voltages?
I think what you're looking for is an understanding of what drives the saturation current of a FET channel. There are various models for this (e.g. short vs long gates, high vs low Vth).
To first order it's electron concentration in the channel and electron drift velocity.
Isat = q * (Ns) * (vs) * W
Ns is number of carriers, vs is saturation velocity.
To second order ohmic contact resistance and other parasitics factor in.
There is some more detailed insight in this (darn interesting, I wish there were an accompanying video!) course presentation:
www.ee.sc.edu/personal/faculty/simin/ELCT563/19%20JFETs%20MESFETs%20HFETs.pdf