I am supposed to be answering this question rather than asking it but I really cannot figure out.

There is a variation on Stone duality linking algebraic and (descriptive) Kripke semantics for (normal only?) modal logics. On the algebraic side, we have modal algebras, i. e. pairs $(B,\Box)$ where $B$ is a Boolean algebra and $\Box$ is a multiplicative operator on it. On the Kripke side we have descriptive Kripke frames, i. e. pairs $(X,R)$ where $X$ is a Stone space and $R$ is a binary relation on $X$ such that $R^{-1}C$ is clopen for any clopen subset $C$ of $X$.

Thanks to this duality there is a whole exciting dictionary translating key concepts and facts between these two worlds. For example, free modal algebras correspond to canonical models, inducing modalities on finite subalgebras of $B$ correspond to inducing relations on finite continuous images of $X$, etc.

There is one really important construction though "on the left" which I have problems translating "to the right side". One frequently needs the following: given an instantiation $\varphi(b_1,...,b_n)$ of a modal formula in $(B,\Box)$, generate finite Boolean subalgebra of $B$ by values of subexpressions of $\varphi$. Although this sounds very syntactic, it can be also made more algebraically flavored: you seek a smallest finite Boolean subalgebra containing the $b_i$ and endowed with a modality (but not necessarily modal subalgebra) in such a way that $\varphi(b_1,...,b_n)$ computed in that subalgebra is the same.

My question is whether there is some recognizable construct on descriptive frames that is dual to the above?

My attempts did not go very far. Clearly specifying $b_1,...,b_n$ means a homomorphism from the free $n$-generated modal algebra to $B$, so dually the initial data consist of an $n$-model, i. e. a p-morphism $\pi:X\to\mathscr K(n)$ from $X$ to the $n$-canonical model, together with a specified clopen $C_\varphi$ of $\mathscr K(n)$ which then produces a clopen $C=\pi^{-1}C_\varphi$ of $X$. From that data we seemingly must produce continuous $R$-preserving map (but not necessarily a p-morphism) from $X$ to some finite Kripke frame $F$ and a p-morphism (?) from $F$ to $\mathscr K(n)$ which factor $\pi$ and give the same $C$, but I am not sure.

**Update**

After discussing the answer below, it now seems that the question reduces to the following.

Let $\mathscr F(n)$ be the free modal algebra on specified free generators $p_1,...,p_n$, and let $\mathscr K(n)$ be its dual (canonical model). Then for any $\varphi(p_1,...,p_n)\in\mathscr F(n)$ we get a uniquely determined finite Boolean subalgebra $B_\varphi\subseteq\mathscr F(n)$ generated by all subexpressions of $\varphi$. Dually this gives a finite quotient $\mathscr K(n)\twoheadrightarrow F_\varphi$ (not necessarily a p-morphism for any choice of $R$ on $F_\varphi$). Then one must describe this $F_\varphi$ explicitly or, even better, characterize it by some universal property.