Dispatching on types with the same UnionAll (but you don't know the types beforehand)

A gist with the code in this post can be found here. When planning a small library in Julia, I kept running into a similar problem across all of the type systems that I tried when looking for the proper abstraction. Briefly, I had a single parametric abstract type at the top of an arbitrary type hierarchy (with any number of abstract and concrete types below it, possibly added later by a user) and I needed to write a function that would only work on objects with the same UnionAll1 types within the hierarchy.