  In a companion paper we showed that the symmetry group $\mathfrak{G}$ of
non-expanding horizons (NEHs) is a 1-dimensional extension of the
Bondi-Metzner-Sachs group $\mathfrak{G}$ at $\mathcal{I}^{+}$. For each
infinitesimal generator of $\mathfrak{G}$, we now define a charge and a flux on
NEHs as well as perturbed NEHs. The procedure uses the covariant phase space
framework in presence of internal null boundaries $\mathcal{N}$. However,
$\mathcal{N}$ is required to be an NEH or a perturbed NEH. Consequently,
charges and fluxes associated with generators of $\mathfrak{G}$ are free of
physically unsatisfactory features that can arise if $\mathcal{N}$ is allowed
to be a general null boundary. In particular, all fluxes vanish if
$\mathcal{N}$ is an NEH, just as one would hope; and fluxes associated with
symmetries representing `time-translations' are positive definite on perturbed
NEHs. These results hold for zero as well as non-zero cosmological constant. In
the asymptotically flat case, as noted in \cite{akkl1}, $\mathcal{I}^\pm$ are
NEHs in the conformally completed space-time but with an extra structure that
reduces $\mathfrak{G}$ to $\mathfrak{B}$. The flux expressions at $\mathcal{N}$
reflect this synergy between NEHs and $\mathcal{I}^{+}$. In a forthcoming
paper, this close relation between NEHs and $\mathcal{I}^{+}$ will be used to
develop gravitational wave tomography, enabling one to deduce horizon dynamics
directly from the waveforms at $\mathcal{I}^{+}$.
