Considering my mounts appear out of thin air at the drop of a hat, and melt when they touch water, I'm not sure they have any actual physiological functions. I mean, your character definitely eats and drinks so clearly bodily functions like blinking happen. I'm not convinced blinking is needed by these horse/cat/dog/camel/robot/skeleton summons.
Life is a ride, like days in a train, cities rush by, like ghosts in the night. The rhythm of wheels, time fades away, stations of a journey, destination unknown.
Chuck Norris added the mounts to the game, now they don't need to eat, sleep or blink anymore.
Also they can still run at full speed without any stamina left.