The session of the superadministrator on the child sites ends

Your logged in status is managed by cookies. Your browser will not send the needed auth cookie if you’re requesting a different domain or even sub-domain. Your browser has no idea that they are of the same network.

You could try one of the single sign-on (SSO) plugins. They work by changing the authentication method so one sign-on is valid for multiple sites.

Thanks! The plugin works, but it is limited. If the transition is carried out in one browser tab, then everything is fine (it slows down a little only). And if you open another network site in a new tab, then the session ends on the previous tab and site. It seems to me that this should not be the case after all, and this is some kind of defect in my installation.

I’m not sure how the plugin actually works. Whatever it is might be hindered by browser security measures. You might try asking the devs through the plugin’s dedicated support forum if this is expected behavior and if anything can be done about it. Sadly, they don’t appear to be responsive on that forum, and the plugin has not been updated in quite a while.

You should consider using a different SSO plugin that’s more actively maintained and supported. I might be wrong, but I don’t think it matters if the plugin is intended for multisite or not. A different domain is a different domain, multisite or not.