Hi @Yihai Zweifel , I assume when you say Move of a ticket you mean a transition between status's.
If you are using Jira Cloud this could be handled via an Automation Rule.
From here as you move the Tickets per your trigger it should load the comment.
Hope this helps.
I should also mention there is an alternate solution here where someone asked a similar question already https://community.atlassian.com/t5/Jira-questions/Automatically-add-a-comment-to-jira-issue/qaq-p/454647
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