  Yang-Baxter integrable vertex models with a generic $\mathbb{Z}_2$-staggering
can be expressed in terms of composite $\mathbb{R}$-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
$\mathbb{K}^\pm$. 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.
