Mastodon supports user migrations that keeps your followers. It's pretty new and not supported everywhere yet, but it does work (I've used it) with the caveat that some followers on older software will still follow the old account at the moment.
It works by the old and new server "cooperating", so it won't let you move between two servers which won't federate with each other - in that specific scenario you'll need to go via a third server which both speak to.
The move appears to you as if your followers unfollows your old account and follows the new account, so they're all "redirected" to your new handle. E.g. you can see that here: https://mastodon.social/@vidarh
You indicate to the old server where you want to move to, and then to the new server where you're moving from, and the fact the old server will specify your new handle in your profile is used as proof when informing your followers instances of the new address.
If you do this, you can see mastodon.social returns a "movedTo" attribute for my old account:
It works by the old and new server "cooperating", so it won't let you move between two servers which won't federate with each other - in that specific scenario you'll need to go via a third server which both speak to.
The move appears to you as if your followers unfollows your old account and follows the new account, so they're all "redirected" to your new handle. E.g. you can see that here: https://mastodon.social/@vidarh
You indicate to the old server where you want to move to, and then to the new server where you're moving from, and the fact the old server will specify your new handle in your profile is used as proof when informing your followers instances of the new address.
If you do this, you can see mastodon.social returns a "movedTo" attribute for my old account: