Hi Michal,
Welcome to the community!
Before diving into your question in detail, I have one question: Are you using a separate (paid) app for managing these 'test cases'? Or are they simply Jira work items like the Epics and stories and tasks? To get some better context, it would really help if you can share a screenshot of such a (dummy) test case, and how they are currently linked to the story/epic.
Having that clarified, and assuming that it's indeed just another work item type in Jira, I think Automation would definitely be the solution for your query. I can help provide suggestions - have you already tried to create a rule yourself? If so, please also share some screenshots of your attempts, so that we can build on that (optional).
Thanks!
Hans
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