Wilkins, 26, hit .293/.338/.538 with 30 home runs and 38 doubles for Triple-A Charlotte in the White Sox system in 2014. He made his major league debut with Chicago to end last season but struggled mightily, going just 6-for-43 (.140) with two doubles, two walks and a whopping 22 strikeouts.
He was a fifth-round pick by the White Sox in 2010.
The Blue Jays claimed Wilkins off waivers on March 29, then optioned him to Triple-A Buffalo, where he was hitting .264/.353/.319, with four doubles and no home runs in 21 games. He was designated for assignment by Toronto on Saturday.
Wilkins has 29 days of major league service time, and is using the first of his three option years this season.
The Dodgers were carrying 13 pitchers, so swapping a position player for a pitcher was inevitable, though it seems unlikely that Wilkins will be the one to stick on the major league roster, at least not immediately.
Baker was 0-1 with a 5.73 ERA in two starts with the Dodgers, allowing seven runs and 11 hits in his 11 innings, with eight strikeouts, three walks and three home runs allowed.
The Dodgers signed Baker to a minor league contract on April 3, and the 33-year-old right-hander had a 1.06 ERA in three starts for Triple-A Oklahoma City, with 16 strikeouts and just one walk in 17 innings.
Baker was originally scheduled to start Thursday's series finale against the Brewers in Milwuakee, but that obviously won't happen now. The odds-on favorite to get the call on Thursday is Mike Bolsinger, who was scratched from his scheduled start for Oklahoma City on Sunday.
Bolsinger and Wilkins were teammates at the University of Arkansas from 2008-2010, and played together in the 2009 College World Series.