Currently, when an issue (sub-task, user story, bug etc) is moved to a new parent it is only possible to configure an automation rule with the "Issue Updated" trigger and you need specific logic to check and see if the parent was changed.

The problem with this is you end up having a high amount of automations being triggered when the chances of it actually doing anything is quite low.

What we'd like to have is utilize the "Issue moved" trigger instead, so that could be fired when the issue moves to a parent which is more logical and also would reduce the amount of times this particular rule is executed. This should be fired in either scenario of moving within the same project or to another project.

See this relevant KB that explains how to set up the Issue Updated trigger for when an issue changes parents: https://confluence.atlassian.com/jirakb/how-to-run-an-automation-for-jira-rule-when-a-sub-tasks-parent-changes-moved-1123194416.html