Bailey, who had to waive his 10-and-5 rights to complete the trade, will reportedly be released by the Dodgers, according to Ken Rosenthal of The Athletic. He's owed $23 million in 2019 - the final season of a $105-million contract signed in 2014 - plus another $5 million to buy out his 2020 club option.
The trade will open up room under the luxury-tax threshold for the Dodgers.
Dealing the mercurial Puig - who was apparently disgruntled by his playing time last season - as well as the veteran Kemp also frees up an outfield spot in Los Angeles. The Dodgers have been connected to free-agent superstar Bryce Harper throughout the offseason.
The Dodgers signed Puig after he defected from Cuba. He quickly became a fan favorite while helping Los Angeles to a pair of NL pennants, despite his overall rocky relationship with the team.
Injuries and platooning by manager Dave Roberts limited Puig to 125 games last year, where he hit .267/.327/.494 with 23 homers, 63 RBIs, and 15 stolen bases while providing Gold Glove-caliber defense in right field. Those numbers were put up primarily against right-handed pitching as he struggled against southpaws; Roberts' decision to use him in a platoon role was what left him upset in the first place.
In Cincinnati, Puig will be reunited with hitting coach Turner Ward, who previously coached him in Los Angeles.
The 28-year-old has one year of team control remaining and is projected by MLB Trade Rumors to earn $11.3 million in 2019.
Wood, who is also entering his final year of arbitration, was expendable after finishing last season as a reliever. The 27-year-old, an All-Star in 2017, put up a 3.68 ERA and 1.21 WHIP with 135 strikeouts in 33 appearances (27 starts) for the Dodgers a year ago.
Wood is expected to be a starter in Cincinnati, according to MLB.com's Jon Morosi.
Kemp is entering the final season of an eight-year, $160-million contract he initially signed with Los Angeles in 2012. This marks the fourth time he's been traded since signing that deal.
The 34-year-old was not expected to suit up for the Dodgers after they re-acquired him from the Atlanta Braves last winter, but Kemp ended up giving them his best season in almost five years. Over 146 games, he hit .290/.338/.481 (121 OPS+) with 21 homers and 85 RBIs, earning his third All-Star berth while winning the Players Choice NL Comeback Player of the Year.
Farmer, 28, spent the majority of last season at Triple-A Oklahoma City, where he posted a .785 OPS with seven homers in 312 plate appearances while splitting time between catcher and third base. He's appeared in 59 big-league games over the past two seasons.
Downs, ranked as the Reds' No. 7 prospect by MLB Pipeline prior to the trade, slashed .257/.351/.402 with 13 homers, 47 RBIs, and 37 steals in 120 games at Single-A Dayton. The 19-year-old shortstop was selected 32nd overall by the Reds in 2017.
Gray, who turned 21 on Friday, was the Reds' second-round pick this past June. The right-hander posted a 2.58 ERA and 0.88 WHIP in 12 starts at Greeneville of the Appalachian League.