Currently the satellites don't have intersat links. There is a ground station in each geofenced region, which makes mobility difficult. I'd expect it to start working once the intersat links are functional.
I don't think that's the issue. Starlink won't even let you move from one cell to another even when the cells are served by the same ground station. And moving from one ground station to another is no harder than handing off from one cell tower to another, which I assume was solved 30 years ago. They just aren't allowing it.
Edit: Maybe not? https://publish.twitter.com/?query=https%3A%2F%2Ftwitter.com... ... though it is already "later this year".