Since upgrading to ArcGIS Pro and therefore the 2.01 version of the API I have noticed that when clone_items is run on a webmap with the "copy_data" set to True and "search_existing_items" set to False then the newly created webmap only respects those rules (i.e. points at cloned copies of the Hosted Feature Layers) for the first item it encounters.
If your webmap contains two different hosted feature layers, it clones them both, and the first instance in the webmap is correctly referenced, but the second one points back at the original hosted feature layer.
This is incredibly frustrating behavior (a halfway house of what it should be doing) and is surely a bug?? Any advice on whether ESRI are aware/fixing it or suggestions for a workaround are gratefully received!!
In case anyone lands here. I gave up and contacted ESRI support. They told me that the 2.01 version of the API contained a lot of bugs for clone_items. I have since upgraded to Pro 3.1.2 and this solves the immediate issue. However, just a warning that items contained in group layers within a cloned webmap still seem to confuse it and point back to the original rather than the clone! I pointed this out in my reply to ESRI so hopefully it will be addressed at some point soon