It is over the plate as it crosses the batter's front armpit and knee, so yes moving in the box moves the strike zone.
Is this the case in all organizations? I was also under the assumption that the zone was based on the front of the plate. Not only up and down, but left and right as well. For example, a screwball thrown at the inside corner would be a strike if it passes the front inside corner of the plate even if the catcher catches it several inches inside. Am I wrong?