  We consider a dynamic protocol for quantum many-body systems, which enables
to study the interplay between unitary Hamiltonian driving and random local
projective measurements. While the unitary dynamics tends to increase
entanglement, local measurements tend to disentangle, thus favoring
decoherence. Close to a quantum transition where the system develops critical
correlations with diverging length scales, the competition of the two drivings
is analyzed within a dynamic scaling framework, allowing us to identify a
regime (dynamic scaling limit) where the two mechanisms develop a nontrivial
interplay. We perform a numerical analysis of this protocol in a
measurement-driven Ising chain, which supports the scaling laws we put forward.
The local measurement process generally tends to suppress quantum correlations,
even in the dynamic scaling limit. The power law of the decay of the quantum
correlations turns out to be enhanced at the quantum transition.
