  For any smooth compact manifold $W$ of dimension at least two we prove that
the classifying spaces of its group of diffeomorphisms which fix a set of $k$
points or $k$ embedded disks (up to permutation) satisfy homology stability.
The same is true for so-called symmetric diffeomorphisms of $W$ connected sum
with $k$ copies of an arbitrary compact smooth manifold $Q$ of the same
dimension. The analogues for mapping class groups as well as other
generalisations will also be proved.
