This question was asked and bountied at MSE, but received no answer.
In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since everything boils down to the number of classes. However, the situation seems more interesting when we restrict attention to continuous reducibility. I'd like to know more about the structure of smooth equivalence relations with respect to continuous reducibility; what is a good source on this topic? (I'm happy to restrict to equivalence relations on Baire space if that would help, but in general I'm interested in arbitrary Polish spaces.)