Fediverse

According to the documentation: Followers, yes; posts, no.

https://docs.joinmastodon.org/user/moving/#export
https://docs.joinmastodon.org/user/moving/#move

There is a migration function in the preferences

As others have said: followers yes, posts, no. Some other Fediverse platforms can migrate posts, though, and I believe Firefish (previously known as Calckey) is able to import posts from Mastodon as well as from other instances of itself.

It's possible to bring your followers with you by doing an account redirect. On your old server and/or account, go to Preferences->Profile and then scroll down until you see move to a new account and click on the link. Then on your new server and/or account go to Preferences->Profile and scroll down until you see move from a different account and click on that link. Your follower will be alerted as to your new account. Right now you aren't able to bring your posts over with you. Unfortunately it is not possible to even export your posts yet with Mastodon. A part of me is regretting not going with Akkoma or Calckey when I decided to redo my Mastodon instance because I believe those twitter-like alternatives give you the ability to export/import all of your data.

Mastodon supports account migration. You can export the list of your followers and the accounts you follow as CSV and import them from another instance. But there's no option to import posts yet.

Also, note that when you migrate all followers in the old account will get automatically unfollowed.

For more details, see https://docs.joinmastodon.org/user/moving