Inbound email with specific actions for an Incident record was received. However, there is no update in the Incident record after the email is received and processed.

Any supported release.

One possible cause for this type of issues is because other inbound action, with no specific conditions, was fired first and having a "Stop Processing" flag set. Therefore, no other inbound action was fired.

Always review the email records and check the log section. 

The log section always tells you what inbound actions were skipped and what were processed as well as if there was a Stop Processing operation. For example, in this case we found: 

2020-07-01 02:49:33 PM Stop processing detected after executing script: Auto Response For Closed Cases

Analyze why the inbound action was fired first, and why it included a Stop Processing action. This may be a design flaw.

As a best practice do always this:

• Review the email log for details of every particular case.
• Check all your inbound actions for conditions, processing order, and 'Stop Processing" flag.
• Analyze what your inbound actions do and what are the chances to have unexpected consequences.
• Use the Order to prioritize execution order. Lower number executes first.

Documentation:

For additional details go to https://docs.servicenow.com and search by "Inbound Email actions"