There should be an option that we can clone live websites onto staging websites within the same client.
Create a new staging website and pick the third option: Clone a Website
xyzulu thanks somehow I overlooked this option!
I had to come here to find this solution, so I would suggest there should be some UI changes so you can also overwrite an existing staging site with the live version, or at least a tooltip to explain how to do this within the CP.
Thanks
