Summary: Let \(r>0\) be a finite delay and \({\mathcal C}_0= {\mathcal C} ([-r,0],H)\) the Banach space of continuous vector-valued functions defined on \([-r,0]\) taking values in a real separable Hilbert space \(H\). This paper is concerned with topological properties of solution sets for functional-differential inclusions of sweeping process type: \[ {du\over dt} \in-N_{K(t)} \bigl( u(t) \bigr)+ F(t,u_t), \] where \(K\) is a \(\gamma\)-Lipschitzean multifunction from \([0,T]\) to the set of nonempty compact convex subsets of \(H\), \(N_{K(t)} (u(t))\) is the normal cone to \(K(t)\) at \(u(t)\) and \(F:[0,T] \times {\mathcal C}_0 \to H\) is an upper semicontinuous convex weakly compact valued multifunction. As an application, we obtain periodic solutions to such functional-differential inclusions, when \(K\) is \(T\)-periodic, i.e. when \(K(0)= K(T)\) with \(T\geq r\).