The issue occurs randomly every day. While using some of the subdomains, the browser will start to wait for a response for long periods. I have checked that this issue is not related to the origin servers. When one of the subdomains hosted on one server does not respond, the other subdomains hosted on other servers (even at a different geographical location) do not respond as well. These subdomains and servers are not dependent on each other, apart from all of them using Cloudflare proxied DNS records. The issue seems to affect only the current browser session. If the same subdomains are visited in incognito mode of the same browser, they respond. The issue resolves/goes away on its own in a few minutes. It seems that issue relates to some domain-level rate-limiting that affect all its subdomains. However, I have checked the traffic and security logs in the Cloudflare dashboard and I have not found any entries related to this. I have furthermore disabled all WAF rules and put the Security level to Essentially Off. Yet, the issue is still present. What is more, I have noticed that when this issue happens, there are also issues with loading the Cloudflare dashboard even when I am logged in to the profile that owns the root domain. Cloudflare would show an error page with a prompt to report a technical issue in those cases.
To sum up, this seems to be related to how Cloudflare handles requests from my local IP and/or how it proxies the traffic for the root domain and its subdomains.
This happens on different computers sharing the same IP address (on the same local network) when using Firefox. I do not know if the issue occurs with other browsers as well.