How to duplicate animated objects and offset the animation?

So I’ve animated a spinning cylinder that also moves up and down, kind of like a drill. I would like to have several spinning cylinders, but offset so they don’t move all in unison.

I’ve duplicated the objects with a group and now have an empty with which I can move them around, that’s working. But I can’t figure out how to offset the animations. Someone mentioned the NLA editor when I googled but I can’t seem to get separate tracks there or anything.

This is what it looks like: https://i.imgur.com/6bSwtCH.png
I’d like to create some small animations, but this is one of the very basic things I’m not sure about at the moment. Would be cool if someone could help.