  A Banach space $X$ is said to have property ($\mu^s$) if every weak$^*$-null
sequence in $X^*$ admits a subsequence such that all of its subsequences are
Ces\`{a}ro convergent to $0$ with respect to the Mackey topology. This is
stronger than the so-called property (K) of Kwapie\'{n}. We prove that property
$(\mu^s)$ 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 $(\mu^s)$ under
$\ell^p$-sums is discussed. For a family $\mathcal{A}$ of relatively weakly
compact subsets of $X$, we consider the weaker property $(\mu_\mathcal{A}^s)$
which only requires uniform convergence on the elements of $\mathcal{A}$, 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 $(\mu_\mathcal{A}^s)$ 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 $L^1(\nu,X)$ (for a finite measure $\nu$) has property
$(\mu_\mathcal{A}^s)$ for the family of all $\delta\mathcal{S}$-sets whenever
$X$ is a subspace of a strongly super weakly compactly generated space.
