It's probably a little dependent upon which plugins are installed/etc.. but the main things are:
DB Changes:
- wp_site: update values for domain and path
- wp_blogs: update values for domain and path for each sub-site/blog
- wp_x_options (one for each sub-site/blog, where x is the blog-id): update values for option_name like 'siteurl', option_name like 'home', option_name like 'upload_url_path' and there'll be others if you've got things like woocommerce, etc. installed, but the user can also update these via the network admin>sites>SITE_NAME>edit>settings themselves..
wp-config.php will need to be updated:
define('DOMAIN_CURRENT_SITE', 'NEWVALUE');
define('PATH_CURRENT_SITE', 'NEWVALUE');
the rest of the multisite lines in wp-config.php should not need to be changed.
AND, seems that many users (myself included) have issues with the .htaccess file, so that might also need to be updated, see here: http://codex.wordpress.org/Multisite_Network_Administration#.htaccess_and_Mod_Rewrite
once you've done that it "should" work, but for me, it's still not - likely something up with the rewrite rules in .htaccess. time for the ametur to step out of the way and pay for a professional!
hope that helps.