The subthreshold swing in mV per decade of the drain current change in the subthreshold region is much affected by the short channel effects as it is a bipolar transistor effect where in the subthreshold region the channel will have a bipolar structure.
As for the change of the subthreshold swing increases with lowering the oxide capacitance. Which means as the thickness of the MOS capacitor decreases its oxide capacitance increases and the the ideality factor decreases leading to a decrease in the threshold swing to smaller values with the minimum of 60 mV/decade.
Please more information about the threshold voltage ID:VDS relationship please refer to the literature in the link: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-012-microelectronic-devices-and-circuits-fall-2009/lecture-notes/MIT6_012F09_lec12.pdf
This one part of the answer of your question.
If you consider that the leakage current across the gate oxide to the source then only the source current will increased by the gate leakage current as
IS= IG+IDS
So only there will be an effective voltage swing for the source current but that of the drain current will be affected by the thickness of the oxide as i explained before.
Generally it is expectorated that that the voltage swing may increase or decrease according to the the dependence of the IG current on VGS.
The subthreshold swing in mV per decade of the drain current change in the subthreshold region is much affected by the short channel effects as it is a bipolar transistor effect where in the subthreshold region the channel will have a bipolar structure.
As for the change of the subthreshold swing increases with lowering the oxide capacitance. Which means as the thickness of the MOS capacitor decreases its oxide capacitance increases and the the ideality factor decreases leading to a decrease in the threshold swing to smaller values with the minimum of 60 mV/decade.
Please more information about the threshold voltage ID:VDS relationship please refer to the literature in the link: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-012-microelectronic-devices-and-circuits-fall-2009/lecture-notes/MIT6_012F09_lec12.pdf
This one part of the answer of your question.
If you consider that the leakage current across the gate oxide to the source then only the source current will increased by the gate leakage current as
IS= IG+IDS
So only there will be an effective voltage swing for the source current but that of the drain current will be affected by the thickness of the oxide as i explained before.
Generally it is expectorated that that the voltage swing may increase or decrease according to the the dependence of the IG current on VGS.