I thought you already had backups of your current installation generated by WordPress Move.
Now, you have two options:
1) Letting WordPress Move handle the transfer of your backup files by properly configuring the plugin.
2) Transferring the backup files yourself.
For the first option, the server you are migrating to needs to have a working FTP server. If you have one running, enter the FTP details of the server you are migrating to, on the WordPress Move configuration page of your current WordPress installation. For the remote backup path, you need to enter the full path to the /wordpress-move/backup/ directory on the server you are migrating to, starting from the top directory that your FTP account has access to. If these sound complicated, follow the instructions on the next paragraph.
For the second option, fill in the configuration page of WordPress Move with incorrect data, so that WordPress Move stops asking you to fill in the fields on the configuration page. Once done, go to Migration Assistant and start the migration. It will be able to create backup files but will fail to connect the target server as we provided incorrect data. Go to Backup Manager and download the backup files by clicking on their names. Open your FTP client and connect to the server you want to migrate to. Then, upload your backup files to the /wp-content/plugins/wordpress-move/backup/ directory on your new server. Once you finish uploading them, you can use the Restore option of the Migration Assistant on the server you are migrating to, to finalize the migration operation.