When creating a reusable SharePoint 2013 workflow from SharePoint Designer using the Ribbon, you are unable to bind it to a content type.
The (very silly) workaround to this is to create the workflow from the File menu as follows.
I think it would be a tough job just by using PowerShell. Basically, you will have to
Do the migration for the subsite.
Get references to the Content Types at the source site collection. Recreate the content types in the destination site collection and then assign them to the same objects in the subsite.
Go through all contents in the subsite and then redo ...
Try to use any third-party migration tool like sharegate, Metalogix etc. Using the tool, first migrate structure of the site, then do the data migration. You can install these tools as a trial free version for some days.
While you are doing the copy structure migration, you can even restructure your site in the target environment, everything is ...
How long have you created new content types? Make sure you have waited 24 hours to make these documents to be crawled for search.
Make sure the documents you want to return with the correct content types.
You could also provide some screenshots with us to understand your issue more correctly.