Transferrring multisite

I have a multisite setup on my old host.
multisite is set up on sub.site.com. My actual site is at site.com. I’m about to add a second site, but there’s no space. I’m moving to another host.

I have a new host at site.net. How can I transfer my old multisite setup to this? If I merely copy all files, and import the DB, then it wouldn’t match the url right? I’ve never tried to import a DB.