The first thing to do is performing an export/import of the database tables used by Responsivizer:
Notice that you could have created several modules assigned to the Responsivizer template, in such case you need to perform an export/import of those records from the Joomla! core tables named: #__modules and #__modules_menu.
All settings of the component are hold in the core database table named #__extensions, you have to search for the record named 'Responsivizer' of type 'component' and migrate it or only the related field named 'params'.
Finally if you created some custom overrides in the template you may need to copy those files as well. Override files are located at: root/templates/responsivizer/html