I think you just need to backup your multisite using something like this tutorial https://www.digitalocean.com/community/tutorials/how-to-migrate-wordpress-from-shared-hosting-to-a-cloud-server-with-zero-downtime
since you are using easy engine you should first set up your droplet and install WordPress as a multisite with subdirectories. then follow the instructions on how to back up your website from the tutorial above.
As an alternative, you can purchase a pro license from servmask to back up your wordpress multisite. you then just install the plugin in on your new easy engine vps instance of wordpress and import the file. I would still backup your project as described in the first tutorial from digital ocean because if anything fails you have the backup files from both the database and the wpcontent folder.
i hope this helps you.