Were you deleting entire history, or non-deformer one? Once you do regular history delete, the character loses it's link to the rig.
And, you should move neither skeleton not body. For animation purposes you'd key animation on those curves e.g. green and yellow circles around the character's torso, which are parts of the rig you are supposed to use.
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Maya2019.1 @ Windows10 & GeForce GTX1080Ti
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