Saves will transfer up automatically! Nothing you need to do. I don't recommend placing new downloads into the old folder - extract it somewhere else (and then you can just delete the old folder entirely).
The Lillie scene specifically re-triggered for everybody since I added a new switch specifically for it. Some folks were able to perform a specific sequence of events that would lock you out of the first cutscene (which would then lock you out of the Drowzee quest entirely). TL:DR is that you seeing the cutscene again broke absolutely nothing - you're fine!