You can use Configuration Manager for JIRA to do that. Moving configuration changes from test to production is one of its main purposes.
If you need help or more information about the product features or a live demo feel free to drop an email to firstname.lastname@example.org
You can probably move the workflow using the workflow export/import available out of the box in JIRA, but this is very unreliable as the workflow itself might have a lot of dependencies like custom fields, roles, etc... and you can't actually "merge" the changes into an existing workflow, but only import it as new and then assign it to the project you want. And at the end this is the solution for just the workflow and the rest should be moved manually.
Yes you need the add-on on both test and production systems. For production work you need to purchase only one license which you will install on production and you can use development license on the test server. The development license is available once you purchase the commercial license.
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