This is exactly what “Backup Project” does. Well, it doesn’t move files, it copies them.
“Save As” can’t do the job as it refers only to the project file (.cpr) and that is why I mentioned above that project files are not necessarily part of the project folder. They are independent of it.
If I just want to play around a bit I always use the same folder as my project folder. I save the project file into that folder (but could save it anywhere else). Once I feel I got something that I would like to work on further I use “Backup Project” and give it its own location. This function will also create a new project file and it will automatically be placed in the new project folder. So all looks nice and tidy afterwards.