I think you need Domain Mapping. Domain mapping allows a blog on a multisite install to serve from any domain name. This way a blog does not have to be a subdirectory of the main install, or a subdomain. The WordPress Default supports Domain Mapping without Alias. Add the Domain in the blog-settings to the blog of the Network administration area.
If you have usaged the search on WPSE, maybe you found this helpful thread 182467. Read it also, is much helpful.
Domain Setting with WP Core
You should add the domains to the site settings of each site in the Multisite.
See the follow example screenshot for a additional domain to other domain with directory structure.
Site Info, Domain Setting
To set a domain, edit the site-settings, tab "Info", also a screenshot, helps much more - I think.
Check also the settings, that the URL works for the site and the home-url.
Often is it helpful - but not necessary, to set the
COOKIE_DOMAIN constant to an empty string in your
define( 'COOKIE_DOMAIN', '' );
Otherwise WordPress will always set it to your network's
$current_site->domain, which could cause issues in some situations.
WordPress Core and Domain Alias
WordPress can handle different domains inside the core, no additional plugin is necessary. A plugin is only important, if you use alias of a domain. If you will use it, then you have two possibilities form ready usable plugins.
- Mercator - WordPress multisite domain mapping for the modern era.
- WordPress MU Domain Mapping - Map any blog/site on a WordPressMU or WordPress 3.X network to an external domain.