When a minor league team changes its major league affiliation, what happens to the players?

I think the players and coaches go to the new affiliate. And the old affiliate retains its name and ownership.

Good question, I’ve never actually thought about that.

Here’s what usually happens:

First off, these changes always happen during the offseason, so it doesn’t really have an effect on the season. The players on the team simply follow their organization to the new city and begin playing for the new team. The old team keeps its name and ownership, as @Tink1113 said.
A real life example: The Columbus Clippers had long been the AAA affiliate of the Yankees. In 2006, the Yankees changed their AAA affiliation to the newly-created Scranton-Wilkes Barre Yankees. So, all of the Clippers team, coaches, and manager shipped out to Pennsylvania and continued their organization there. The ownership of the Clippers was left to recruit a new MLB affiliate, which they found in the Washington Nationals. So, the Nationals’ AAA affiliate packed their bags and moved to Columbus.
This happens quite frequently in the minors, so there is little continuity in the franchises.