Hello,
I have quite some experience with Wordpress sites, so I the job will be done fast and easy :)
Does your new hosting provide all the needed prerequisites for those Wordpress sites?
- php (version?);
- mysql (version?);
- apache (version?).
How many databases you have in your new hosting?
What versions are your Wordpress sites?
Are your nameservers updated for the new hosting?
Regards,
Ovidiu