Along with placing Turner on the DL retroactive to Monday, the Dodgers optioned pitchers Mike Bolsinger and Zach Lee to Triple-A Oklahoma City and activated Thursday pitching acquisitions Alex Wood, Jim Johnson and Luis Avilan.
Mat Latos, also acquired Thursday, will be activated before his Sunday start, when a corresponding move will be necessary.
Morse, 33, was designated for assignment after being acquired in Thursday's deal with Miami, which was built around bringing Latos to the Dodgers for Minor League pitchers Jeff Brigham, Victor Araujo and Kevin Guzman.