think that specific kinds of motion produced by canter that you described have many effects. One is proprioception, vestibular, and cerebellar sensorimotor integration. Another is to work on the sacral, parasympathetic plexus of autonomic nerves that sends safety signals to the brainstem via the vagus nerve.
We did a lot of work on cortisol and purinergic signaling early on. The release of extracellular ATP is a direct trigger of cortisol release from the adrenal cortex. This is a fundamental part of the cell danger response (CDR). You’ve hit on a hugely successful way of helping to turn off this danger/stress signal through healthy, rhythmic motion. Antipurinergic drugs like suramin have a similar action. They relieve stress and anxiety at a cellular level and allow the world to open up. The sky is the limit for a child who is no longer blocked by societal misconceptions and cellular stress and fear.