Human body-parts do not fall asleep as the prescribed notion but they tend to lose the sensation of touch or pressure due to the compression of the nerves, which takes the stimulus from the body to the brain. Mostly, it is because of the position of our limb that defines the sleepiness of our body-part. It is the nerve which is compressed and, therefore, the distal part of the compression gives us no sensation, also referred to as sleepiness here.
Nerve fibres take the message to the brain and bring it back. When we move ourselves or change the position, the nerve releases and then we feel tingling, burning before we feel our sensations coming back to normal and numbness disappearing. The ability to move and touch comes immediately after we have gained control of the limb minus the numbness of the nerves, which bring the sensation to touch and position. This sensation is connected to the brain via the spinal cord. When we move our body, it is back to normal.