I have code bases that are about to become significantly shorter because of both. I do need to sit down and perform an analysis to understand the performance implications of this, thought I suspect there aren't many.
I'm not sure what to make of local functions. I've read and re-read about them but I still don't see the utility over a private function, unless they were intended to alleviate the performance issues that come with using a local Func<> inside of a method.