The "falling asleep" sensation that we all experience from time to time is medically known as a parasthesias. Often described as a "pins and needles" experience, this happens because of transient damage to a nerve (often from compression). This is normal as we can all compress a nerve occasionally depending on our body position. However, as you suggest, if it is happening persistently and happening regardless of position, this could be a sign of a nerve problem. I would recommend that you see your doctor
to have this evaluated.
A problem with the nerves is known as a neuropathy. This requires evaluation. There are many causes of neuropathy, but if it is localized to only the right foot -- this would suggest that it is a mechanical problem of the nerve that supplies the right foot being pinched. There are multiple places where the nerve can be pinched (starting from the back where the nerve exits the spinal cord all the way down the foot itself). Your doctor can evaluate this. If the nerve is being pinched, physical therapy or potentially surgery
may be required to help this.
There are also some causes of diffuse nerve injury
-- often that affect the feet first. Diabetes, thyroid problems, vitamin problems are even some infections can cause this.
Talk to your doctor to have this evaluated. Good luck!