Two issues are treated in this work. (i) The generic fact that, if a fermionic superfluid in the BCS regime overflows from a narrow container into a much wider one, pairing is much suppressed at the overflow point. Physical examples where this feature may play an important role are discussed. (ii) A Thomas-Fermi approach to inhomogeneous superfluid Fermi systems is presented and shown to work well in cases where the local density approximation breaks down.