  Consider a proper, isometric action by a unimodular locally compact group $G$
on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. Then an
equivariant Dirac-type operator $D$ on $M$ under a suitable boundary condition
has an equivariant index $\operatorname{index}_G(D)$ in the $K$-theory of the
reduced group $C^*$-algebra $C^*_rG$ of $G$. This is a common generalisation of
the Baum-Connes analytic assembly map and the (equivariant)
Atiyah-Patodi-Singer index. In part I of this series, a numerical index
$\operatorname{index}_g(D)$ was defined for an element $g \in G$, in terms of a
parametrix of $D$ and a trace associated to $g$. An Atiyah-Patodi-Singer type
index formula was obtained for this index. In this paper, we show that, under
certain conditions, $\tau_g(\operatorname{index}_G(D)) =
\operatorname{index}_g(D)$, for a trace $\tau_g$ defined by the orbital
integral over the conjugacy class of $g$. This implies that the index theorem
from part I yields information about the $K$-theoretic index
$\operatorname{index}_G(D)$. It also shows that $\operatorname{index}_g(D)$ is
a homotopy-invariant quantity.
