To my surprise and delight I was able to do this. I figured I should document it somewhere. It synthesizes a couple other answers I found to guide me but focuses on a case that I haven't seen addressed directly.
This is my situation: a couple years ago I replaced my home computer. My old computer was running Ubuntu 14. My new one is running Ubuntu 16. On the old computer, I was making regular backups to an external drive using backups/deja-dup/duplicity.
A couple days ago, the hard drive on my old computer (Ubuntu 14 one) died. I had copied over most the important stuff to my new computer (Ubuntu 16). However, there was a projects folder that still had some older projects that I hadn't gotten around to copying over. So I wanted to see if I could restore this folder from the backups on the external drive.
This is the path of the folder from my old computer I wanted to restore:
This is the path of the folder on my external drive where the backups were being saved (after plugging external drive into new computer):
I wanted to restore it to a folder at this path on my new computer: