If I, sitting at a workstation on my network, move data from one remote drive to another remote drive, does any of it reside at any point on my local machine?
Most likely yes. Assuming you initiated the transfer on the workstation, and the remote filesystems are both mounted on the workstation, and the copy is mediated by the workstation. You should expect that as a minimum a filesystem block worth of data is buffered. In reality the copy may buffer much more data, as it can usually improve performance quite a bit. But the ability of the system to do that depends upon the particular network file system you are using.
In an enterprise level system this might not be the case. A Fibrechannel switch (or the director within) connecting many disks can usually mediate data transfers itself, keeping data off the controlling system. It will depend on lots of questions exactly what happens.
Yes, if you copy a file (files) from disk A to disk B, most copy programs (windows, DOS window, etc., Linux??) will read in sequence a buffer-full of A’s data, then write it out to disk B. If A or B or both are remote disks shared on the network, the data will travel across the network to your PC, either on the way in, the way out, or both.
Does any of it it end up on your local disk? Not likely. Systems like Windows and Linux can pretend they have more memory than they do; if they need too much, they swap the least-used parts back to disk (C:\pagefile.sys, in Windows). However a full-blast copy is probably using the memeory quickly and often enough that it never becomes dormant enough to be “swapped out”… Unless your computer is so busy full that it spends most of its time swapping between the programs it is trying to run, and everything slows to a crawl.
Many systems are smart enough, though, that if you move (not copy) from one directory/folder to another on the same remote drive, rather than rewrite the file, the disk system just moves the pointer to that file from one directory to the other.
And yes, some storage subsystems (high end disks attached to network or server) are smart enough to recognize such move requests and handle them off-line - but I would assume that would require your PC to have the correct drivers for that subsystem - to recognize the request as “Hey, this can be done by the subsystem, I don’t have to read the entire data contents one bit at a time.” and pass the simple move request to the subsystem.