I have a simple project automation for sending email upon support request creations, that is working fine, but there are cases when users move ticket from one project to another, is it possible to set automation in that case to send email notification ?
I don't understand what you want do do completely.
You want to send a notification when an issue is moved from one project to another right?
In Automation this would be the issue moved event. It's only available in the full version, not in the lite version. I'm not sure which one you have.
You'll find a comparison on this page https://confluence.atlassian.com/automation/getting-the-right-version-993924597.html
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