  We study the adiabatic pumping of the Chern-Simons axion (CSA) coupling along
a parametric loop characterized by a non-zero second Chern number $C^{(2)}$
from the viewpoint of the hybrid Wannier representation, in which the Wannier
charge centers (WCCs) are visualized as sheets defined over a projected 2D
Brillouin zone. We derive a new formula for the CSA coupling, expressing it as
an integral involving Berry curvatures and potentials defined on the WCC
sheets. We show that a loop characterized by a non-zero $C^{(2)}$ requires a
series of sheet-touching events at which $2\pi$ quanta of Berry curvature are
passed from sheet to sheet, in such a way that $e^2/h$ units of CSA coupling
are pumped by a lattice vector by the end of the cycle. We illustrate these
behaviors via explicit calculations on a model tight-binding Hamiltonian and
discuss their implications.
