Can you elaborate on your process/intended results and how you make a copy of the layer a bit more?
I would say export the data as a new feature class after each near analysis, however you have done that and say another near analysis overwrites it, which I wouldn't expect at all - the copied layer is not referenced in the tool parameters at all?
Ensure you run the processes completely separately then do a series of table joins at the end. I'd also advise generating a new ID field to ensure you can correctly join back the features at the end, as sometimes OBJECTIDs can get regenerated.