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For integers g and n, let V^{2n+1}_{g} denote the g-fold boundary connect-sum of the manifold D^{n+1}\times S^{n}. In this talk I will discuss recent work of mine with Boris Botvinnik where we determine the (co)homology of the classifying spaces BDiff(V^{2n+1}_{g}, D^{2n}) in the direct limit as g approached infinity, in the case that 2n +1 > 7. In particular we identify its homological type with the infinite loopspace QBO(2n+1)< n >. This calculation enables the determination of all characteristic classes for fibre bundles with fibre V^{2n+1}_{g}, for large g. This result can be viewed as an analogue of the Madsen-Weiss theorem for high dimensional manifolds with boundary.