Heyward has been activated from the 10-day injured list due to a back tightness, and he is projected to be available for Friday's game against the Reds.
Hurt has been sent to the 60-day injured list with right shoulder inflammation, and he is not expected to be activated until at least the beginning of June.
Kershaw is expected to remain on the 60-day injured list until at least the end of July while he recovers from surgery to repair the gleno-humeral ligaments and capsule in his left shoulder.
May has been shuttled to the 60-day injured list after having surgery to repair his flexor and UCL in his right elbow, and he is not likely to suit up until the end of July.
Cavalli has been placed on the 60-day injured list while recovering from Tommy John surgery, and he is not expected to make his season debut until sometime in June.