I have searched, but haven't found anything about this. Is there a feature/add-on available that would automatically change the state of an issue after a defined amount of time. For example, I want an issue to move from a "Done" state to "Assigned" after it has been sitting in Done for 5 days. And every time an issue with that particular type gets moved into the Done state the counter starts again, ultimately ending with it going back to the Assigned state.

Workaround

Use Automation with a scheduled trigger as discussed here:
https://www.atlassian.com/software/jira/automation-template-library/auto-close-jira-issues
As with all Automation rules, be aware of your Automation service limits