Some local network properties connected with convergence of sequences are preserved by countable box-products. For example, if topological spaces $X_n$, $n\in\omega$, have countable $cs$-networks at points $x_n\in X_n$, then the box-product $\square_{n\in\omega}X_n$ has a countable $cs$-network at the point $(x_n)_{n\in\omega}$.
We recall that a family $\mathcal N$ of subsets of a topological space $X$ is a $cs$-network at a point $x\in X$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ convergent to $x$ and any neighborhood $O_x\subset X$ of $x$ there exists a set $N\in\mathcal N$ such that $N\subset O_x$ and $N$ contains all but finitely many elements of the sequence $(x_n)$.
Another property preserved by countable box-product is the existence of an $\omega^\omega$-base at each point.
We recall that a topological space $X$ has an $\omega^\omega$-base at a point $x\in X$ if there exists a neighborhood base $(U_\alpha)_{\alpha\in\omega^\omega}$ at $x$ indexed by elements of $\omega^\omega$ so that $U_\alpha\subset U_\beta$ for any $\beta\le\alpha$ in $\omega^\omega$.
Spaces with $\omega^\omega$-base are extensively studied in this paper and often appear in Topological Algebra and Functional Analysis.