  In this work we consider natural generalizations of local complementation in
Banach spaces, which include Lipschitz-local complementation. We show that all
these notions are indeed equivalent to the classical notion of local
complementation of Banach spaces. As an application, we show that local
complementation is naturally preserved under certain free constructions in
Functional Analysis, including Lipschitz-free spaces and free Banach lattices.
