An MRI on Wednesday determined Young had a strained right forearm, and Meden has right rotator cuff inflammation. Young's stint is retroactive to Tuesday and Medlen's to Wednesday.
Manager Ned Yost said neither injury was severe and that both players should be OK to return when their time on the DL is over.
The left-handed Duffy, who mostly was a starter before becoming a reliever this year, will begin on a 50-pitch limit.
Young is 1-5 with a 6.68 ERA in seven starts and Medlen is 1/3 with a 7.77 ERA in six.
Moylan is 1-1 with a five saves and an 0.71 ERA in 12 relief appearances with the Storm Chasers, and Alexander is 1-0 with a 2.57 ERA in 14 innings.