If you want to create a setup so when a user logs out of a child site, they are redirected to the homepage of the site, one easy-to-use plugin solution is the WP Logout Redirect plugin. After the plugin has been installed and activated on the child site, it has no settings to be changed. An example would be if you used a membership plugin or WooCommerce, then you needed to have a customer or user redirect on logout back to the homepage, and not stay on the existing page.
If you need a plugin which has more options for setting custom URL redirects by user role for example, then Peter’s Login Redirect plugin would be of use.
Once the plugin has been installed and activated, you can set custom URL logout redirects from:
Login/logout redirects Settings
The Peter’s Login Redirects plugin will add one custom database table, which is named;
wp_login_redirects
You can also use the code snippets extension from MainWP to set a code snippet on all connected child sites, using this specific key.
You now have two plugin options, as well as a code snippet option to set a homepage redirect on your child site.