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.

If it doesn't update the old content on your old account, then is there a way to copy the old content to your new account?

I imagine that is what ggg-parent meant by: "it does involve down- and uploading and/or copypasting zips/datafiles."

Indeed.

Though it also involves migrating blocklist, followers, some settings, and such.

Yeah, this is how it works. I don't know the technical details/internal terminology, but I've definitely had a few accounts I follow switch servers for various reasons. It was basically just "oh hey so-and-so's handle changed" from my perspective, which is kinda neat.