Important note: If the Theme being used originated from the Shopify Theme Store, then a valid purchase of that Theme is also required on each account for a valid license, in order to use that Theme.
As such, the destination store will also need to purchase a valid license for that Theme as well.
Without a valid license, merchants are not able to receive support for the Theme from the respective Theme developer.
For your Theme, Replay is able to automatically copy the Theme itself, its coding, its asset levels, and the assets directory. If Media/Image Files were uploaded into the Theme via the Files section of your store - as found via the Document/Files Uploader within your Store Settings - Replay will be unable to copy those. Shopify does not yet allow access to this Files section.
However, if your Theme contains Media/Image Files from the Files section of your Store Settings, then once the Theme has been copied, Replay will automatically email you with a confirmation of completion, as well as a ZIP file containing all of the Files the app located within your copied Theme.
From there, you can simply upload all of the Files from that folder directly into the Files section of your Store Settings! :)