If a client sends a REPLY to a case email (which is from an "email to case" email address), can SF not be told to send a copy of the email and/or send an alert to the Case Owner? This seems no-brainer to me, but I'm not finding help for it (Google, Community, etc). Otherwise, a Case Owner would need to check the case periodically manually to find out if any replies have come – this seems silly.
From Setup > Customize > Cases > Email-to-Case, check Notify Case Owners on New Emails.
Or if that's not working, in the meantime you could trigger EmailMessage to create a CaseComment instead, then enable Case Comment Notifications from Setup > Support Settings.