Powerful confluence criteria for higher-order rewriting exist for left-linear systems, even in the presence of critical pairs and non-termination. On the other hand, confluence criteria that allow for mixing non-termination and non-left-linearity are either extremely limited or hardly usable in practice. In this paper, we study confluence criteria which explore sort information to make proving higher-order confluence possible, even in the presence of non-termination and non-left-linearity. We give many interesting examples of systems covered by our results, including a (confluent) variant of Klop’s counterexample, and a calculus issuing from a dependent type theory with cumulative universes.