The White Sox got veteran relievers Luis Avilan and Joakim Soria and cash. The Dodgers also obtained minor league infielder Jake Peter, and the Royals added a pair of minor leaguers in the deal Thursday.
The 28-year-old Alexander emerged as a key pitcher for the Royals last season, going 5-4 with four saves and a 2.48 ERA in 58 games. The Dodgers lost durable Brandon Morrow from their bullpen when he signed with the Chicago Cubs, and lefty Tony Watson became a free agent.
Avilan was 2-3 with a 2.93 ERA in 61 games last season for the Dodgers. The 28-year-old lefty struck out 52 in 46 innings.
Soria was 4-3 with one save and a 3.70 ERA in 59 games for the Royals last season. The 33-year-old righty has 204 career saves.
The 24-year-old Peter hit a combined .279 at Double-A and Triple-A in the White Sox organization.
Kansas City got 24-year-old starter Trevor Oaks, a righty who went 4-3 with 3.83 ERA at Triple-A for the Dodgers, and 23-year-old infielder Erick Mejia, a switch-hitter who spent most of the season at Double-A for Los Angeles, batting .278 with 28 stolen bases overall in the minors.