Sort of. You can move your list of follows, and you can automatically inform your followers that you've moved and most of the time, but not all of the time, their instance will automatically update their list of follows, but you can't automatically move your content or backdate it on the new instance.
Is that a design descision, or just something that's not implemented?
Adding an 'originally-posted' field to exports, and allowing mass-posting on the target should technically work, and non-technically be a tricky problem.
> Is that a design descision, or just something that's not implemented?
The Mastodon API (currently the defacto API for Fedi clients) has no support for backdated posts (see [1]) and servers will push out any status you create to your followers which would mean a deluge of previously seen content. Backdating and a flag for "do not push out backdated posts" could be added[2] but there's a whole raft of problems that come along with it (resource usage for large accounts, who can use the functionality, how do you stop abuse, etc.)