The client would have to fetch all your messages first. Some rooms can have thousands of messages a day, and some people are in thousands of rooms [1] (though these are core developers so they might be stress testing)

[1]: https://news.ycombinator.com/item?id=32021303

but you would hopefully not export all messages every time? incremental?

If you wanted automated exports I think an easier way would be to host your own homeserver, and create your account on that homeserver. I believe any messages in any rooms you join will be synced with your homeserver, and you can perform automated full backups (or maybe incremental backups if the database supports it?)