  Given a closed smooth manifold $M$ of even dimension $2n\ge6$ with finite
fundamental group, we show that the classifying space ${\rm BDiff}(M)$ of the
diffeomorphism group of $M$ is of finite type and has finitely generated
homotopy groups in every degree. We also prove a variant of this result for
manifolds with boundary and deduce that the space of smooth embeddings of a
compact submanifold $N\subset M$ of arbitrary codimension into $M$ has finitely
generated higher homotopy groups based at the inclusion, provided the
fundamental group of the complement is finite.
