How to export Multisite database to an .sql file (no cPanel/phpMyAdmin)

I have a WordPress Multisite installation, with 210 blogs, and I need to move it to a different server. The one that the websites are on does not have cPanel / phpMyAdmin, and the new one does.

I know that I need to copy the database to the new location, and also all the WordPress files to the new server.

I was going to export the database files to an .sql file, and import it to the new server with phpMyAdmin. I was looking at the plugins for doing that, but none of them (as far as I was looking) are for WordPress Multisite. I could get sql files, but for 210 files, and this is obviously not what I am looking for.

Is there any way to do it?

Thank you.