The direct answer is: by layering the controls.
I am not familiar with this rig, so I will reverse-engineer it in my head.
Let's assume, all of those small ball-shape beige controllers are from the level #1, those control fine details (mouth shape and antennae smaller circles also are from this layer). Click one of those controllers, open the Outliner, press F. Explore the node hierarchy up and down.
Most likely, you will notice each of those is parented to some auxiliary object (group/locator) to zero out transforms, which in turn is parented to some SplineIK chain, that is cluster-deformed by the larger green curve, which belongs to a level #2 controls, that control larger mesh volumes (red antennae rings, green tail ball etc are from this layer too). So, when you lift the green curve, it makes bell shape for that underlying curve, that transforms the joints to which the beige controls are parented/constrained; same joints bend the main body mass.
The #3 level are yellow curves, except the outer ring, that allow to do rough body transition through the scene.
The #4 level is the outer yellow ring, that has master controls and is meant for positioning the character in the scene.
The level numbering can be reveresed and even more subdivided, it just was more easier to explain starting from the minute, toward the general.
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Maya2019.1 @ Windows10 & GeForce GTX1080Ti
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