Some local network or base properties are preserved by countable box-products of topological spaces. 
In particular:

1) the existence of a countable $cs$-network at each point;

2) the existence of a countable $cs^*$-network at each point;

3) the existence of a countable $s^*$-network at each point;

4) the existence of an $\omega^\omega$-base at each point.

Now I recall the corresponding definitions.

Let $X$ be a topological space and $x$ be a point of $X$. A family $\mathcal F$ of subsets of $X$ is called a 

$\bullet$ a *$cs$-network* at $x$ 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 $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains all but finitely many elements of the sequence $(x_n)$;

$\bullet$ a *$cs^*$-network* at $x$ 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 $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains infinitely many elements of the sequence $(x_n)$;

$\bullet$ an *$s^*$-network* at $x$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ accumulating at $x$  and any neighborhood $O_x\subset X$ of $x$ there exists a set $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains infinitely many elements of the sequence $(x_n)$;

$\bullet$ an *$\omega^\omega$-base* at $x$ if each set $F\in\mathcal F$ is a neighborhood of $x$ and $\mathcal F$ can be written as $\mathcal F=\{F_\alpha\}_{\alpha\in\omega^\omega}$ so that $F_\alpha\subset F_\beta$ for any elements $\beta\le\alpha$ of $\omega^\omega$ (endowed with the ccordinatewise partial order).


Those notions are studied in [this paper][1] and often appear in Topological Algebra and Functional Analysis. 

For a topological space $X$ and a point $x\in X$ we have the implications:

($X$ has an $\omega^\omega$-base at $x$) $\Rightarrow$ ($X$ has a countable $s^*$-network at $x$) $\Rightarrow$ 

$\Rightarrow$ ($X$ has a countable $cs^*$-network at $x$) $\Leftrightarrow$ ($X$ has a countable $cs$-network at $x$).

  [1]: https://arxiv.org/abs/1607.07978