On Thursday, the Coyotes signed Kuemper to a two-year contract extension.
"The Coyotes have a young and talented team and a very bright future ahead," Kuemper said in a team statement. "I'm excited to be a part of it and can't wait to contribute."
The 27-year-old Kuemper was 10-1-3 with a 2.10 goals-against average in 19 games for the Kings this season. He's 51-35-17 with a 2.52 GAA in 104 career games with Minnesota and Los Angeles.
The 6-foot-5 Kuemper will join Antti Raanta in goal for the Coyotes.
"Darcy is a big, talented goaltender who is having an excellent year," general manager John Chayka said in a team statement. "You need great goaltending in this league in order to be successful and with Antti and Darcy, we are confident that we have an excellent tandem for the future."
The 25-year-old Rieder, from Germany, had eight goals and 11 assists in 58 games for Arizona this season. He has 51 goals and 60 assists in 292 career games, all with the Coyotes.
"We continue to look for opportunities to improve our team speed and Tobias will bring that dynamic to our club," Kings general manager Rob Blake said in a team statement. "We appreciate Darcy Kuemper's contributions this season as he has played very well for us in his role."
Wedgewood was 5-9-4 with a 3.45 GAA in 20 games this season for Arizona. The 25-year-old goalie appeared in four games for New Jersey in 2015-16.