Some good news about #Mastodon, there has been a very significant bit of code added which will automatically fetch all the replies in a thread. This means everyone will be able to see a thread's whole conversation no matter which server they are on!
The code has just been "merged" which means it should become available in the next update to Mastodon, hopefully within the next few months.
(Those who want the technical details can see the Pull Request at https://github.com/mastodon/mastodon/pull/32615)
(Via @rolle)
You can already move accounts (https://fedi.tips/transferring-your-mastodon-account-to-another-server/).
If you mean moving posts, the posts remain on your old server and redirect people to your new account.
Moving Mastodon posts was attempted by Firefish but the moved posts were clones of the originals and they broke all of the discussion threads on the originals. Also the moving process overwhelmed the server as it involved potentially thousands of posts being added to the server at once from a single account.