  A quantum many-body system undergoes phase transitions of distinct species
with variations of local and global parameters. We propose a framework in which
a dynamical quantity can change its behavior for quenches across global
(coarse-grained criterion) or local system parameters (fine-grained criterion),
revealing the global transition points. We illustrate our technique by
employing the long-range extended Ising model in the presence of a transverse
magnetic field. We report that by distinguishing between algebraic and
exponential scaling of the total correlation in the steady state, one can
identify the first transition point that conventional indicators such as the
rate function fail to detect. To determine the second one, we exploit the
traditional local quenches. During quenches with and without crossing the
critical points along the local parameter, total correlation follows either the
same or different scaling laws depending on its global phase.
