Migrating domain name from old WP host to new (which uses vhosts)

You can set the domain for the WP install in the config-file wp-config.php and so its easy to set an new domain for a install.

define('WP_SITEURL', 'http://www.example.com');
define('WP_HOME', 'http://www.example.com/blog');

Now use WordPress this domains and the settings in backend was not deciding.

But the task after this small change is, that you change the url to the new domain in the database, like for attachments or images. She are inside the tables, with completly link. You can use the plugin Search & Replace or you change directly in the database via a db-tool, like Adminer or phpMyAdmin.