  Introducing internal degrees of freedom in the description of topological
insulators has led to a myriad of theoretical and experimental advances. Of
particular interest are the effects of periodic perturbations, either in time
or space, as they considerably enrich the variety of electronic responses, with
examples such as Thouless's charge pump and its higher dimensional cousins, or,
higher-order topological insulators. Here, we develop a semiclassical approach
to transport and accumulation of general spinor degrees of freedom, such as
physical spin, valley, or atomic orbits, in adiabatically driven, weakly
inhomogeneous insulators of dimensions one, two and three under external
electromagnetic fields. Specifically, we focus on physical spins and derive the
spin current and density up to third order in the spatio-temporal modulations
of the system. We, then, relate these contributions to geometrical and
topological objects -- the spin-Chern fluxes and numbers -- defined over the
higher-dimensional phase-space of the system, i.e., its combined
momentum-position-time coordinates. Furthermore, we provide a connection
between our semiclassical analysis and the modern theory of multipole moments
by introducing spin analogues of the electric dipole, quadrupole and octapole
moments. The results are showcased in concrete tight-binding models where the
induced responses are calculated analytically.
