What URL are you using?
You should use a URL like this in your email template to open the current case from an email:
The user will be forced to login if there is no open login session, and then it will redirect to open the case.
If the use is currently assigned to an open task in the case, then the user will be able to edit the case. If the user isn't assigned to a currently open task in the case, then ProcessMaker will redirect to the case summary (which is the same as what you see under Home > Participated when opening a case).