Maybe I misunderstand something here. I assumed we were talking about work to electrode voltage and further than that, the divided one.
So apart from the ignition or ohmic sensing the cutting itself is nothing dramatic, well at least to me its not. What do you mean by the "nanosecond" rates?
So purely from work to electrode voltage point of view the plasma does not "instantaneously" turn on the cutting power. The current is ramped up in the tens or hundreds ms time frame. This is all depending on the torch/consumable design. Same goes when you stop cutting, the arc is "extinguished" in a controlled manner.
Typically the IGBT's switch around 50kHZ, I have seen some a bit higher, some a bit lower but all that I came across switch under 100kHZ. The arc itself I see as a purely resistive load.
Now for crossing a kerf, the arc lags behind for a few degrees, depending on speed and material. When you cross a kerf, the top of the arc will touch the material before the bottom and makes the arc "jump" over the kerf. This extends the arc to a certain degree and makes the arc voltage higher (longer arc). Something similar happens when the center slug drops and the arc follows the slug for a moment, extending the arc.
Anyone can test their plasma's max output voltage during cutting by incrementing the cut height and seeing when the arc goes off. A plasma rated for example 200V cannot sustain an arc that requires 300V.
I will try to remember to scope the kerf crossing and hole cutting next time I have a chance, maybe not this year but for sure start of 25.
I'm eager to learn new things and it would be great if someone corrects me if I am wrong.