Yes. The process has three steps: First, you update the receiving profile to "allow" the move by pointing at each other using the as:alsoKnownAs relationship. This allows everybody to confirm that the receiving account is participating in the move, and authorizes it. Then, you update your old account with the "movedTo" property, so that any new users who look up your account will see a notification that you've moved. Finally, you send out a Move activity to all of your followers, pointing at the new account. Automatically, all of your old followers who receive the Move activity will send Follow activities to the new "receiving" profile.
This process doesn't update any of the old content from your account, which was regarded at the time as a necessary simplification because of the issues of updating canonical URIs for accounts on one system to accounts on another system (different software might have different expected routes, you might need to store lookup tables, etc etc. It just opens up a huge can of worms). In practice this doesn't really matter that much since Mastodon is used primarily for microblogging and less for, well, actual blogging. If you were designing a more fully-featured social blogging platform like a Medium or Tumblr equivalent you'd probably want to put some more thought into that side of things.