Washington are sending off guard De'Angelo Russell, a future second-round pick, and a future second-round pick swap to the Memphis Grizzlies.
The six-team deal involved the Bucks and Pistons trade from earlier that sent Caris LeVert to Milwaukee.
Middleton, 34, signed a three-year, $17.6 million deal as part of the sign-and-trade deal.
He started the first half of this past season with the Wizards before the team traded him to the Dallas Mavericks. The vet, now entering his 15th season, was originally dealt to the Wizards during the 2024-2025 season in a trade with the Milwaukee Bucks.
Middleton finished this past season averaging 10.2 points, 3.7 rebounds, and 2.8 assists.
After spending his rookie year with the Pistons, Middleton spent the next 11-and-a-half years with the Bucks, where he earned three All-Star selections and was a major piece in the team's 2021 championship run.