  In this paper we consider systems of vectors in a Hilbert space $\mathcal{H}$
of the form $\{g_{jk}: j \in J, \, k\in K\}\subset \mathcal{H}$ where $J$ and
$K$ are countable sets of indices. We find conditions under which the local
reconstruction properties of such a system extend to global stable recovery
properties on the whole space. As a particular case, we obtain new
local-to-global results for systems of type $\{A^ng\}_{g\in\mathcal{G},0\leq
n\leq L }$ arising in the dynamical sampling problem.
