If it is really a multisite and you’re moving sites to a new multisite then NS Cloner might be your best bet…
https://wordpress.org/plugins/ns-cloner-site-copier/
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This may sound weird in a way but it’s the way I do this…
Create a WordPress instance that follows the same layout you created originally. You’ll want the exact same directory structure.
FTP is your friend here if you have it as you can go back and FTP in your media, themes, and plugins folders from the wp_content directory. Else use the control panel’s File Manager.
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Once you have your new WordPress working save a copy of your wp_config.php then take the database credentials section of the old site and copy that into the new site’s wp_config.php replacing what’s there in your current (new site’s) database credentials section…
You’ll probably need to point the database address from ‘localhost’ to the actual address of your MySQL database as provided by your old webhost.
Backup everything in your old site…
Write down the current IP address of your DNS entry for the main domain then point your main domain name at your new host…
Save that wp_config you modified in the new site now and, once the DNS records propagate, your site should start working with the old database server. If it crashes due to a database error you may need to tweak the access settings at the old sites CP to allow outside access.
You’ll probably see some media file errors. You’ll need to FTP in and save those files if you didn’t do that originally.
Correct any problems then point your other domains at the new host… Test each as you go for issues, correct those before moving on to the next.
Once you have everything working install a backup plugin and backup the database…
Revert to the new site’s saved original wp_config.php get the backup plugin working and restore the database backup.
Test everything once more!
You should now have everything running on your new host.
Let me know if you need further clarification.
As I mentioned, NS Cloner makes moving the sites easy but my method allows you to test while moving things incrementally.
My method also allows you to run your database on one server and your site(s) on another server which helps when your servers are loaded heavily.
