Why do you need a processed folder? The image files are not changed. All of the adjustments are the catalog. You could create a collection for your processed work and then export your copies to a folder of your choice and add it to the catalog. Maybe I'm not understanding you..
I moved processed pics to a separate folder so that I can easily select all of them to export to JPG's. This also tells me what pictures I have reviewed as I always process chronologically.
You could do that with a collection. As I indicated, moving "finished" images doesn't really accomplish anything because there are no changes applied to the image file. You are simply moving unmodified image files from one folder to another. But if it works for you then go for it. All of the changes are in the catalog. As long as the catalog knows where you moved the images you'll be fine.
The "why" is because the folders are empty. LR is not a file browser; it only knows about folders that contain images that have been imported.
In general, it is best not to move files after they have been imported.
My folder structure for each photo shoot has two folders, "originals" and "exports". I tend to process the images in chronological order. The history panel for each image tells me if the image has been edited or not. I also use the star ratings; less than three stars does not get edited.