  Recently we pointed out the so-called Local Time Scheme as a novel approach
to quantum foundations that solves the preferred pointer-basis problem. In this
paper we introduce and analyze in depth a rather non-standard dynamical map
that is imposed by the scheme. On one hand, the map does not allow for
introducing a properly defined generator of the evolution nor does it represent
a quantum channel. On the other hand, the map is linear, positive, trace
preserving and unital as well as completely positive, but is not divisible and
therefore non-Markovian. Nevertheless, we provide quantitative criteria for
dynamical emergence of time-coarse-grained Markovianity, for exact dynamics of
an open system, as well as for operationally-defined approximation of a closed
or open many-particle system. A closed system never reaches a steady state,
while an open system may reach a unique steady state given by the L\" uders-von
Neumann formula; where the smaller the open system, the faster a steady state
is attained. These generic findings extend the standard open quantum systems
theory and substantially tackle certain cosmological issues.
