It is not meant for you to move folders into the project, it is meant for you to use the migrate tool and select the content folder of the new project. Chances are you are not mirroring the folder structure exactly so it is looking in the wrong place.
If you want to take the migration somewhere empty first, I would recommend instead migrating to a bran new blank project and using that as a staging ground. That is what I do anyways since it gives you a chance to clean up any extra nonsense that may have tagged along and resave. I will keep migrating to blank projects until only what I need comes over.