  We obtain constraints on the topology of families of smooth $4$-manifolds
arising from a finite dimensional approximation of the families Seiberg-Witten
monopole map. Amongst other results these constraints include a families
generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$
theorem. As an application we construct examples of continuous
$\mathbb{Z}_p$-actions for any odd prime $p$, which can not be realised
smoothly. As a second application we show that the inclusion of the group of
diffeomorphisms into the group of homeomorphisms is not a weak homotopy
equivalence for any compact, smooth, simply-connected indefinite $4$-manifold
with signature of absolute value greater than $8$.
