Will Ukrainian airstrikes inside Russia shift the war?

Possibly. They will make it harder, a lot harder for the Russians to take or advance on Kharkiv further, which is the second largest city in Ukraine, millions of people near the front lines. And if the Russians were to take it or destroy it, level it, you'd have millions of refugees that would be streaming out and into neighboring countries. Not something anyone in NATO wants to see. That is what is less likely to happen, because the Ukrainians can now hit Russian targeting outside of Ukraine.