It is the general instability of joints in Unity, would happen also with normal ragdolls without any PuppetMaster.
Please check out the tips on improving joint and ragdoll stability in Unity, see if any of those help.
Also, scene view handles are actually quite snappy, when you move them by mouse, their motion is not smooth. If you moved the target via code it will already be much smoother. You can also calculate the delta motion of the target and add so of it to each ragdoll velocity in FixedUpdate to transfer some of that acceleration directly to the ragdoll, making it less violent.