  We investigate the evolution of quantum information under Pauli measurement
circuits. We focus on the case of one- and two-dimensional systems, which are
relevant to the recently introduced Floquet topological codes. We define local
reversibility in context of measurement circuits, which allows us to treat
finite depth measurement circuits on a similar footing to finite depth unitary
circuits. In contrast to the unitary case, a finite depth locally reversible
measurement circuit can implement a translation in one dimension. A locally
reversible measurement circuit in two dimensions may also induce a flow of
logical information along the boundary. We introduce "measurement quantum
cellular automata" which unifies these ideas and define an index in one
dimension to characterize the flow of logical operators. We find a
$\mathbb{Z}_2$ bulk invariant for two-dimensional Floquet topological codes
which indicates an obstruction to having a trivial boundary. We prove that the
Hastings-Haah honeycomb code belongs to a class with such obstruction, which
means that any boundary must have either nonlocal dynamics, period doubled, or
admits anomalous boundary flow of quantum information.
