Regarding the implementation of CopyAsync it will always copy the stream.
Can previous claims be applied to the copy as well?
Your current implementation exposes two sort of operations:
They are sharing the same signature (more or less). The override functionality is not controllable in case of Move from the consumer perspective.
The Copy ...
Errors aren't answers
return new DriveInfo(path).DriveType == DriveType.Network;
will harm and not help you. What if a caller passes the integer 3 for path? The network drive status is certainly not True, but you can't say that it's False, either. It's an error, and ...