Hello! My son has epilepsy and this sounds similar to seizures he used to have. They were labeled "Complex-partial seizures". He would lose focus on what was going on, and stare off with only his eyes blinking repeatedly, almost rhythmically. Sometimes he would feel stiff, and other times he would not. These happened at the same time of day as well. Usually within an hour of waking in the morning and within an hour of waking from naps. Sometimes he was responsive to me while I was talking to him, but he couldn't avert his gaze to look at me. Other times he was not responsive to me. Just some food for thought! I hope everything has worked out well. Smiles!
It is likely a benign condition called transient tic disorder, and such involuntary muscle movements are often exacerbated by fatigue or other physical stress. However, his parents should consult with the pediatrician about this and not assume anything.