  The assumption of local relaxation in inhomogeneous quantum quenches allows
to compute asymptotically the expectation value of local observables via
hydrodynamic arguments known as generalized hydrodynamics (GHD). In this work
we address formally the question of when an observable is ``local enough'' to
be described by GHD using the playground of partitioning protocols and
non-interacting time evolution. We show that any state evolving under a
quadratic Hamiltonian can be described via a set of decoupled dynamical fields
such that one of those fields can be identified with a space-time-dependent
generalisation of the root density. By studying the contribution to a connected
spin correlation of each of those fields independently, we derive the locality
conditions under which an observable can be described using the root density
only. That shows both the regime of validity for hydrodynamic approaches that
aim at describing the asymptotic value of observables in term of the root
density only, such as GHD, and the locality conditions necessary for
Gaussianification to occur.
