A Banach space X is said to have property ($$\mu ^s$$ μs ) if every weak$$^*$$ ∗ -null sequence in $$X^*$$ X∗ admits a subsequence, such that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapień. We prove that property $$(\mu ^s)$$ (μs) holds for every subspace of a Banach space which is strongly generated by an operator with Banach–Saks adjoint (e.g., a strongly super weakly compactly generated space). The stability of property $$(\mu ^s)$$ (μs) under $$\ell ^p$$ ℓp -sums is discussed. For a family $$\mathcal {A}$$ A of relatively weakly compact subsets of X, we consider the weaker property $$(\mu _\mathcal {A}^s)$$ (μAs) which only requires uniform convergence on the elements of $$\mathcal {A}$$ A , and we give some applications to Banach lattices and Lebesgue–Bochner spaces. We show that every Banach lattice with order continuous norm and weak unit has property $$(\mu _\mathcal {A}^s)$$ (μAs) for the family of all L-weakly compact sets. This sharpens a result of de Pagter, Dodds, and Sukochev. On the other hand, we prove that $$L^1(\nu ,X)$$ L1(ν,X) (for a finite measure $$\nu $$ ν ) has property $$(\mu _\mathcal {A}^s)$$ (μAs) for the family of all $$\delta \mathcal {S}$$ δS -sets whenever X is a subspace of a strongly super weakly compactly generated space.