Feature request: import/export API of messages

This issue has been migrated from [#3716](https://github.com/matrix-org/synapse/issues/3716).

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Would be cool to import/export (unencrypted) messages from/to Synapse with an API.

An usecase is importing messages from other systems or would be also good to move users from a "testing" Synapse server, which becomes unfortunately "semi-productive", to a new Synapse installation. With a better usernames than "testuser1", "testuser2"...

I read...

https://matrix.org/docs/guides/client-server.html
https://github.com/matrix-org/synapse/tree/master/docs/admin_api

... but couldn't find any API.

Thank you.