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's 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.
Tabata had been outrighted to Triple-A Indianapolis, where he was hitting .291 in 44 games after hitting .289 in 27 games for the Pirates earlier in the year. Tabata is paid $4 million this year, $4.5 million next year, with options for 2017-20. He is expected to be sent to the Minor Leagues.