When I need to move large numbers of files I do it in the operating system and then in LR I re-associate (NOT re-import) the images. This is easy when all your images are under one parent folder as yours appear to be (Our Stuff or MyPictures). It also has the advantage of moving any files in the same folders that are not in the catalog. These are the steps.
1. Go to your catalog settings and choose to backup next time you quit LR and then quit LR to create a recent backup of your catalog. Also ensure you have a proper backup (or two) of the image files. Remember that the catalog does NOT contain your images and a backup of the catalog is NOT a backup of your image files.
2. With LR not running copy the parent folder (Our Stuff or MyPictures in your example) to it's new location. Now you will have the files in the same folder structure (that's important) in both locations.
3. Now start LR again. If the parent folder is not showing right click on any folder and choose "show parent folder". Once the parent folder is showing right click on it and choose "Update folder location" and choose the new location of the parent folder. This should re-associate that folder and the contents of all sub folders. You should see the question mark go away.
4. Rename to original parent folder in the operating system. This will break any remaining links LR may be holding. If you have properly re-associated your files then no ? icons should appear on your folders.
Once you are sure all your images are properly associated to the new location you can delete the original parent folder.
If something does go wrong you can always return to your backup catalog.