  In this paper we prove a homological stability theorem for the diffeomorphism
groups of high dimensional manifolds with boundary, with respect to forming the
boundary connected sum with the product $D^{p+1}\times S^{q}$ for $|q - p| <
\min\{p, q\} - 2$. In a recent joint paper with Boris Botvinnik (see
arXiv:1509.03359 ), we identify the homology of $colim_{g\to
\infty}BDiff((D^{n+1}\times S^{n})^{\natural g}, \; D^{2n})$ with that of the
infinite loopspace $Q_{0}BO(2n+1)\langle n\rangle_{+}$, in the case that $n
\geq 4$. Combining this "stable homology" calculation with this paper's
homological stability theorem enables one to compute the (co)homology groups of
$BDiff((D^{n+1}\times S^{n})^{\natural g}, D^{2n})$ in degrees $k \leq
\tfrac{1}{2}(g - 4)$. This leads to the determination of the characteristic
classes in degrees $k \leq \tfrac{1}{2}(g - 4)$ for all smooth fibre-bundles
with fibre diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$.
