Hi, i think that there we may have 2 ways:
1. Ideal current source in drain.
There are transistor in saturation until Vgs > Vth, because any changes in Vgs > Vth doesn't change drain current (ideal source). If Vgs goes below Vth we have open circuit and therefore no current.
2. Non-ideal current source (transistor current source, current mirror etc)
For this situation current of mosfet in saturation:
\[I=\frac{K}{2}\cdot{(Vgs-Vth)}^{2}\cdot(1+\lambda * Vds)\],
hence if current is constant and if Vgs goes to low, then Vds must goes to high to provide constant current.
If so, from some moment our current source begins to give smaller current.