  Previous works by Gor\'e, Postniece and Tiu have provided sound and cut-free
complete proof systems for modal logics extended with path axioms using the
formalism of nested sequent. Our aim is to provide (i) a constructive
cut-elimination procedure and (ii) alternative modular formulations for these
systems. We present our methodology to achieve these two goals on a subclass of
path axioms, namely quasi-transitivity axioms.
