Here is an interesting post about the Kursk invasion:
A large article in the Kyiv Post:
https://www.kyivpost.com/opinion/37215.
And another opinion about the Kursk invasion. This one is by Eugene Onegin, who I think has general anarchist leanings.
The bold Ukrainian incursion into Kursk has several probable strategic objectives. The disruption of enemy supply lines, and the diversion of Russian military resources to border control are the most immediate ones. Digging in and holding Russian territory as leverage in negotiations that may happen after the US Presidential election is less likely, but by no means impossible.
The fourth goal that comes to my mind is one that I haven't seen discussed elsewhere. The counter-invasion is in fact a demonstration for the benefit of Ukraine's Western allies that yet another one of Putin's 'red lines' has been a bluff all along. Thus far, Ukraine has had to fight this war under diplomatic restrictions that are, as far as I know, completely unprecedented in the entire history of warfare. The aggressor may strike wherever he pleases, whereas the defender is limited to only striking the aggressor on the defender's own internationally recognized territory. A completely impossible way of fighting a war. This restriction was eased in a very limited way quite recently, allowing Ukraine to return fire at Russian artillery positions right behind the border, in order to make Kharkiv defensible. But as I said, this was a very limited exception.
What Ukraine is perhaps doing here first and foremost is asserting its right to defend itself by destroying the enemy's supply lines, depots, airfields, staging areas, troops, etc, etc, before they cross the border, while showing their Western allies that doing so with Western-supplied weapons may make Putin cry and soil his pants, it does not in fact constitute any kind of nuclear 'red line.'