How to Navigate a New Company in a Private Equity Setting

What's unique or especially hard about doing this with a private equity fund? I'd say there's maybe two different ways to answer that. If you swap out even one member of an executive team, it's a totally new team. How do you navigate those waters? And where are those original loyalties? Is it to the founders, is it to the new CEO? Does the new CEO know enough about the context to really be successful?