DropBox

Are you saving files from drive a to dropbox then downloading to drive b from dropbox?

Are both drives at same location?

Doing it via dropbox you are limited buy up and download speed of your web connection.

Not my preferred method, but depend on your situation as little info has been given.

 

Why would you think copying them out to an Internet storage site would be faster?

What do you mean by "a 1TB remote HD"? Where does that HD sit? Remote usually means a different physical location.

If all the drives are physically in one place and connected to your computer, drag and drop is one way but I'd look to use something like CarbonCopyCloner or a similar tool to clone the file structure over and validate that the copy was successful.

If your old drives are USB2 this is going to be a slow operation no matter what. Start it up and go do something else for a few hours.

--- Post Merged, Nov 18, 2016 ---

Also, while you're of course welcome to post questions wherever you like, you may find more people familiar with the Mac Pro over on subforum specific to that model rather than this one which focuses on the Macbook model.

Mac Pro subforum: http://forums.macrumors.com/forums/mac-pro.1/