You distinguish between the List and Set monad; you could also consider the Multi-Set monad, which records how many times a given element is contained but otherwise does not care about the order.

Are you are aware of the paper by Mark Jones on monad transformers? He defines a function swap which gives a condition for when a monad can be lifted into another monad.

The other reference you might not be aware of is the book by Barr & Wells: Toposes Triples and Theories. IIRC monad transformers are covered in Chapter 9 Section 2. As an exercise you can show that this approach is the same as Jones’.