  A system having macroscopic patches in different topological phases have no
well-defined global topological invariant. To treat such a case, the quantities
labeling different areas of the sample according to their topological state are
used, dubbed local topological markers. Here we study their dynamics. We
concentrate on two quantities, namely local Chern marker and on-site charge
induced by an applied magnetic field. We demonstrate that the time-dependent
local Chern marker is much more non-local object than equilibrium one.
Surprisingly, in large samples driven out of equilibrium, it leads to a simple
description of the local Chern marker's dynamics by a local continuity
equation. Also, we argue that the connection between the local Chern marker and
magnetic-field induced charge known in static holds out of equilibrium in some
experimentally relevant systems as well. This gives a clear physical
description of the marker's evolution and provides a simple recipe for
experimental estimation of the topological marker's value.
