I would suggest to use Search and Replace utility when you move your WordPress Network. it seamlessly update all the fields of the database from old domain to new domain. After that, you only need to update the domain name in wp-config.php.
(Tip: do not include http:// and trailing slash while replacing the domain rather just the domain name e.g. www.site.com to www.newsite.com)
Other than the above procedure, three tables are has the domain settings for WordPress Network.
- wp_blogs
- wp_site
- wp_options (for each site wp_XX_options)
Please verify the above tables and DOMAIN_CURRENT_SITE constant in wp-config.php.
Absolute URLs should also be updated (in all posts, pages and widgets) to link to new domain.