JohnO I assume you mean you have asset management included in your license right?
I don’t think you can copy sideways, but you can copy down from processes on Configuration Item level for sure. The reason you cant is because processes built at the lower level use status, conditions, actions, defined at that level that wouldn’t exist if you copied between levels. However copy down from the parent Configuration Item level means that child objects inherit Actions, Conditions, Statuses from the parent.
Anyway easy to do a quick test to demonstrate one way or another.