But it's a slow and smooth motion without any focus breathing intended to highlight an object or an actor, not just autofocus hunting to find something
A dedicated person, called a focus puller, has a remote with a wheel on the side. By turning this wheel the focus puller can remotely control the focus ring of the lens.
The remote usually allows the focus puller to set the maximum range of motion with A/B points. The system doesn't automatically execute the focus pull, but with the hard stops at the A/B points the focus puller can make sure they don't overshoot the target.
Back then the focus puller would use a remote connected by a cable, which is why you see a focus puller running behind the steadicam op in that video.
That's why while you've got A/B points set and marks set on the focus control, you still need a human.
Focus can be controlled manually on the iPhone (in 3rd party apps).
Depending on various conditions (lighting, lens choice, etc) there might be a very large distance range that is in focus - or it might just be a few inches. Even if the focus puller isn’t doing any big focus swings, they are likely making small adjustments.