The Hogwarts Express leaves from Platform 9 3/4 at London's King's Cross Station. Its existence was overlapped with the real Muggle parts of the station, as well as Muggle objects, etc. This could only be possible if the Hogwarts Express existed in a parallel dimension.
On the other hand, its destination -- Hogwarts -- was in a real world dimension (it could be reached with flying cars etc). So, for the Hogwarts Express to travel to its destination, it would have to be a dimension-crossing transport vehicle. Am I right? Or have I assumed something wrong?