Given an ascending chain $(I_n)_{n\in\mathbb{N}}$ of $\Sym$-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals $(I_n^\vee)_{n\in\mathbb{N}}$. Using a novel combinatorial tool, which we call \emph{avoidance up to symmetry}, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of $I_n^\vee$ is given by a polynomial in $n$ for sufficiently large $n$. The same is true for the number of orbit generators of minimal degree, this degree being a linear function in $n$ eventually. The former result implies that the number of $\Sym$-orbits of primary components of $I_n$ grows polynomially in $n$ for large $n$. As another application, we show that, for each $i\geq 0$, the number of $i$-dimensional faces of the associated Stanley-Reisner complexes of $I_n$ is also given by a polynomial in $n$ for large $n$.