Yes, Coupon Codes, Promo Codes are part of Promotions. When you move Promotions using StagingPro, StagingPro follows the following steps:
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As of now all promotions are moved in its entirety. Moving selective promotions will be considered as part of the development roadmap.
IMPORTANT: Be advised that Promotions moved through UI cannot be edited. Workaround is to delete the promotion, recreate it correctly in source and then migrate so it transfers across correctly.