Matrix won’t necessarily download all state/messages automatically, but if your client requests a non-available message your matrix server will query other matrix servers for it.

E.g. if you scroll up to older messages, it might take a a few seconds but your client should eventually show them.

Matrix server use a back-off for servers sending messages, so if your server is offline for many hours, it might take a day for your servers to get messages pushed to by other servers again.