I'm using JIRA Server v7.2.1 and JIRA Service Desk v3.2.1. We have a Service Desk project set up with an automation rule set up such that when a linked issue is transitioned and the resolution is not empty, add a comment to check the related issue for more details.
When the linked issue is transitioned from In Progress to Closed (with a Resolution of "Complete"), nothing happens to the Service Desk issue.
If the same issue is reopened or transitioned to any other status, then a comment is added to the Service Desk issue.
And the comment itself varies on what it indicates as the status of the linked issue. For instance, while writing this question, I'm testing it out with two Service Desk issues (duplicates) that are linked to one JIRA issue. I transitioned the JIRA issue to Closed (Complete), and nothing happened on either Service Desk issue. When I moved them back to Open, both Service Desk issues received a comment that the status of the JIRA issue had changed. One of the Service Desk issues said that the new status was Closed, and the other Service Desk issue said that the new status was Open. Both of these comments were added due to the single event of me changing the status back to Open.
The expected behavior is that the linked issue would transition completely and then the automation rules would run so that it can check the rules against the new status.
In case it makes any difference, I have the rule set up to run as the user who triggered the rule.