I would assume any insurance related issues Fox has would follow Tucker wherever he goes.
That said, I have heard that CNN's new boss's plan is to move right a bit. Might be able to price all that in if he hires Tucker. I don't think it would work though.
Who was that lady that Fox fired a few years ago. She did her best to take a bite out of Trump in a debate. The one who insisted on air that Santa is white. She got a job at like CBS or something after fox and bombed hugely. Once you go that far right, you can't convincingly soften.