In Saturday's exchange, the Brewers dealt infielder Cutter Dykstra -- the son of former New York Mets and Philadelphia Phillies outfielder Lenny Dykstra -- plus cash considerations to the Nats.
Morgan compiled a .253 average with 34 stolen bases and 24 RBI in 136 games for Washington last season, but he drew a pair of suspensions late in the year, totaling 15 games.
He threw a ball into the seats at Philadelphia in August and used "inappropriate language" toward a fan in a Sept. 1 game against Florida, during which he also had an altercation with Marlins pitcher Chris Volstad.
A seven-game suspension for the Philadelphia incident was later called off.
Morgan, 30, hit .351 over 49 games in 2009 after the Nats acquired him from Pittsburgh. But he broke bone in his hand, cutting short a promising season.
Dykstra, 21, hit .312 with five homers, 39 RBI and 27 stolen bases over 100 games last year in Class-A.