So you'd have a quad that would just hover, and you'd have a second radio that would manipulate servos to make the arms and legs go up and down. You'd need a sufficiently powerful quad (probably a hexacopter) to lift the secondary battery and motors to wind the strings (servos wouldn't have enough throw to make the arms move properly).
I don't think it's quite possible YET, only because I don't think you could make it light enough for what you're asking it to do. You'd need weight at the ends of the arms and legs to pull the puppet down when you reverse the motors, and the rest of it would have to be out of foamboard or lightweight plastic or paper, and you'd need some guides to keep the strings from jumping off of your winder motor spools, thus adding more weight...It might be possible now with what we have, but for the weight and mechanics, I don't think you'd have very long flight times (maybe a minute or two, TOPS).
I guess I would have to build it to get the idea across. The head-quad doesn't just hover. It moves-sways around and the body follows.
The body, though almost full size, need not be heavy if made from flat foam board. I am not suggesting a complex 3D shape. That keeps the quad to a reasonable size and allows it to fly for a reasonable length of time.
You could make it a string-puppet with servos to pull the strings, but that is not what I am suggesting. I will explain by describing just one arm.
The flat plate upper arm hangs from the shoulder via a pin joint. The flat plate forearm and hand connect to the upper arm with a pin joint at the elbow. A tiny Gremlin size motor and prop are attached to the hand where the thumb would go. When the motor powers up it pulls the forearm up so it pivots at the elbow. When the elbow gets to 90 degrees it has a stop so it won't bend any more, and then the whole arm comes up until the hand is at about shoulder height, and then the shoulder joint hits the stop. There is no need for strings to pull up anything as the motor and prop on the arm do the pulling.
A similar motor and prop just above the knee pulls the leg forward and then up as it pivots at the hip and knee joints. 4 of these little gremlin motors don't weight much, nor does the battery or ESC required to power them. A RX in the body controls these 4 motors, so you don't need any servos. Precise motion like a robot is not possible, but arms and legs bouncing around like dancing is.
There is no need to add weights to anything. Gravity and the down wash from the quad should pull everything back down just fine.
You do need 2 "Pilots" one for the quad, and one for the body. The quad pilot just sways around with the music. The body pilot just moves the arms and legs around with the music. Some clever mixing in the TX can allow the two sticks to control the 4 motors in a coordinated way that makes the movement look more natural, but ultimately making puppets move is a skill and an art that takes a bit of practice.