The deal also sends forward Patrick Baldwin and a second-round 2025 NBA Draft pick while the Wizards also obtained guard AJ Johnson and a first-round pick swap in 2028.
More moves league-wide are likely ahead of Thursday's trade deadline, with Phoenix benching Kevin Durant for Wednesday's game at Oklahoma City with a left ankle sprain.
That sparked talk Durant is the subject of trade talks with great speculation he could return to the Golden State Warriors, with whom he won NBA titles in 2017 and 2018.
The Bucks, 26-22 and on a three-game losing skid, made the swap with the NBA-worst Wizards (8-41) to land forward Kuzma, who has averaged 15.2 points, 5.8 rebounds and 2.5 assists this season.
Middleton helped the Bucks win the NBA crown in 2021 and had been a mainstay of the Bucks since coming over in a 2013 trade after spending his rookie campaign in Detroit.
He underwent off-season surgeries on both ankles and did not make his debut this season until December 6th.
For his career, Middleton has averaged 16.7 points, 4.8 rebounds, 4.0 assists and 1.2 steals a game.
Other reported deals on Wednesday sent German power forward Daniel Theis from New Orleans to Oklahoma City and Boston Celtics reserve guard Jaden Springer to the Houston Rockets.