They can definitely impeach him after he leaves office. He can be convicted and precluded from ever running for office again. That’s really the only purpose of doing something stupid like this.
You know about Alcee Hastings, right.
Alcee was impeached (and convicted!) as a Federal Judge and then became (and still is!) a Democratic member of Congress.
So, yeah, Alcee can’t be a federal judge again... :-)
He’s a private citizen when he leaves office. Where does the Constitution say a private citizen can be impeached? This doesn’t make sense.