Yang-Baxter integrable vertex models with a generic Z 2 -staggering can be expressed in terms of composite R -matrices given in terms of the elementary R-matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices K ± . We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang-Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases.