It is very easy to migrate WordPress site to another hosting automatically without FTP client, you just need to get Backup & Restore Dropbox PRO plugin.
Follow the guidance to migrate website correctly:
1) Login at your new WordPress website dashboard – that is your target domain TO which you want to migrate;
2) Upload PRO plugin version like it is shown in this guidance: http://www.wpadm.com/faq/install-backup-restore-dropbox-pro-at-wordpress/
3) Connect to your Dropbox Account where you have the backup of your old website FROM which you want to migrate: http://www.wpadm.com/faq/connect-dropbox-full-backup-plugin-to-dropbox-api-v2/
You will not see any backups
4) To display backups, open plugin Settings and tick Read all backup from dropbox cloud.
5) All the backups of your old website will appear in your plugin.
6)Click on Restore in front of the backup you want to restore.
In appeared window write the address of your old domain and click Restore.
7) When migration is done successfully, there will pop new window to sign in. Use login credential from your OLD domain (FROM which you migrated)
8) Reload the page to login to your newly migrated website. Use login details from your OLD domain