Do African Grey Parrots stay on the move?

yes, in search of food. they are known to fly over 50 miles a day in search of food, or better known as foraging for food. that is why when they are "pets", you are supposed to teach them how to hunt for their food, or hide it for them to find. specific "foraging toys" are made for this purpose. since in the wild they fly miles to find food, when in a cage they need something similar to occupy their time, hence the foraging toys so they literally don't go insane of boredom.