Many systems are involved in moving the limbs.
The musculo-skeletal system (muscles and bones) is typically seen as "the" system responsible for limb movement.
However, muscles won't contract unless they're innervated by the nervous system, and won't be able to perform the complex reactions needed for movement unless they're provided substances like oxygen (through blood) from the cardiovascular system. Additionally, muscles are affected by hormones and other chemicals in the body that are released by the endocrine system (they might signal growth, for example).
For further reading, you may want to look up Actin and Myosin and examine muscle contraction.
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