When E-mail based approval is not working, please verify following points -
• Check whether you have configured E-mail notification with proper template where required Macro is in place for state movement
• Check whether Approval E-mail has reached the mailbox and have got attached to the record
• Check whether approval E-mail attached to the record has short code as part of E-mail Subject, if not then update the short code in State names. Please note that short code is must for approvals.
• Check the Person who has approved the record over the E-mail has access to do the state movement in workflow, some time due to multiple Approval roles you might have not selected the proper role to do state movement. (In this case you will get clear error message in history saying you do not have access to do this state movement.)
• Check if approval manager of the record is clicking on Approved button in the E-mail which he has received or is he just replying to the mail saying approved, if he is replying saying approved then state movement will not happen. Approval manager is supposed to click on approved button in E-mail received which opens up new Email window which needs to be sent by approval manager.
If you still see the state movement is not happening post checking all the above points, do reach out to SapphireIMS support team for further debugging.