We work in $\sf ZFC+GCH$. Let $D$ be a class of uncountable regular cardinals, and for every $\alpha$ let $\Bbb Q\_\alpha$ be either trivial if $\alpha\notin D$, or forcing with these two properties:


1. $|\Bbb Q\_\alpha|=\alpha$, and
2. $\Bbb Q\_\alpha$ is $\alpha$-distributive (any less than $\alpha$ dense open sets have a dense intersection).


Let $\Bbb P\_\alpha$ be the finite support product of $\Bbb Q\_\beta$ for $\beta<\alpha$. 



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> Suppose that $\alpha\in D$, is there are condition which implies that $\Bbb P\_\alpha$ preserves the truth value of "$\Bbb Q\_\alpha$ does not add bounded sets to $\alpha$"?
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The condition can be made on either the forcings themselves, or on $D$ (e.g. $(\sup D\cap\alpha)^+<\alpha$ or something like that).



Preserving distributivity with finite support products
GCH implies that $\mathbb{Q}\_\alpha$ is $\alpha$-distributive in the generic extension by $\mathbb{P}\_\alpha$. 


Note that $\Vdash\_{\mathbb{Q}\_\alpha} ``\check{\mathbb{P}}\_\alpha$ is $\check\alpha$.c.c.$"$. For successor cardinal $\alpha$ this is clear, as $|\mathbb{P}\_\alpha| < \alpha$. For strongly inaccessible $\alpha$, the strong inaccessibility of $\alpha$ is preserved under $\mathbb{Q}\_\alpha$. Therefore, was can apply the usual $\Delta$-system argument in the generic extension by $\mathbb{Q}\_\alpha$ and conclude that $\mathbb{P}\_\alpha$ is still $\alpha$-c.c. (and even $\alpha$-Knaster).


Let $\dot{\tau}$ be a $\mathbb{P}\_\alpha \times \mathbb{Q}\_\alpha$-name for a new set of ordinals of size $\mu$ for some $\mu <\alpha$. Let $G\subseteq \mathbb{Q}\_\alpha$ be a generic filter. In $V[G]$, $\mathbb{P}\_\alpha$ is $\alpha$.c.c. and $\alpha$ is still regular. Therefore, there is a $\mathbb{P}\_\alpha$-name $\dot{\sigma}\in V[G]$, $$V[G] \models |\dot{\sigma}| = \nu < \alpha,\ \text{and }1\_{\mathbb{P}\_\alpha}\Vdash\_{\mathbb{P}\_\alpha} \dot{\tau} = \dot{\sigma}.$$ 


But $\mathbb{Q}\_\alpha$ is $\alpha$-distributive, and therefore $\dot{\sigma}\in V$. We conclude that there is a condition $q\in \mathbb{Q}\_\alpha$ such that $(1\_{\mathbb{P}\_\alpha}, q) \Vdash \dot{\tau} = \dot{\sigma}$ and in particular $(1\_{\mathbb{P}\_\alpha}, q)\Vdash \dot{\tau}\in V^{\mathbb{P}\_\alpha}$.


