Where are the files that control the multisite user dashboard?

Most of these files are in wp/admin/network. But be aware that some are just wrappers for regular admin files one level up: they include the corresponding file.

What you probably need are the hooks and the function is_network_admin() that checks if you are currently on a network admin page.

Most hooks and global variables are the same as in admin just with a prefix or suffix network. $pagenow on the dashboard for example is dashboard-network.

The most important difference: there is no network settings API. You can use wp-admin/admin-post.php as a workaround (example).