Here's how to clone a group and all its contained nodes:
layer.add( anyContainerGroup.clone({draggable:true} );
To make your draggable container groups, do the same thing you did with the small rects
Make your blue, green & gold stroked containers un-draggable (they are "originals")
Make a clone of each container placed directly on top of the original (the clones are draggable)
When a small rect is dropped in the clone container, add the small rect to that clone container
The clone containers (including any contained small rects) can be dragged to your dropzone.