  Generalized Kauer moves are local moves of multiple edges in a Brauer graph
that yield derived equivalences between Brauer graph algebras of multiplicity
identically 1. Moreover, these derived equivalences are given by a tilting
mutation. The goal of this paper is to generalize this result first for Brauer
graph algebras with arbitrary multiplicity and second for a generalization of
Brauer graph algebras called skew Brauer graph algebras. In these contexts, we
prove that the generalized Kauer moves induce derived equivalences via tilting
mutations. We also show that skew Brauer graph algebras of multiplicity
identically 1 can be seen as the trivial extension of skew gentle algebras.
