Let $\mathcal{E}$ be a complete and cocomplete category. Given a functor $i: \mathcal{C} \to \mathcal{D}$ between small categories, there is a triple of adjoint functors between their respective categories of presheaves with values in $\mathcal{E}$: $$Lan_i =: i_!: [\mathcal{C}, \mathcal{E}] \to [\mathcal{D}, \mathcal{E}]$$ $$(-)\circ i =: i^\ast: [\mathcal{D}, \mathcal{E}] \to [\mathcal{C}, \mathcal{E}]$$ $$Ran_i =: i_\ast: [\mathcal{C}, \mathcal{E}] \to [\mathcal{D}, \mathcal{E}]$$
Suppose we are further given functors $p: \mathcal{C} \to \mathcal{C}$ and $q: \mathcal{D} \to \mathcal{D}$ such that $i \circ p = q \circ i$. It is then obvious that $p^\ast \circ i^\ast\simeq i^\ast \circ q^\ast$. Using the unit $\mathrm{id}_\mathcal{C} \to i^\ast \circ i_!$ and counit $i_! \circ i^\ast \to \mathrm{id}_\mathcal{D}$ of the first adjunction, we obtain a natural transformation of functors $\mathcal{C} \to \mathcal{D}$: $$i_! \circ p^\ast \to i_! \circ p^\ast \circ i^\ast \circ i_! \simeq i_! \circ i^\ast \circ q^\ast \circ i_! \to q^\ast \circ i_!$$ Is this a natural isomorphism in general? What about if $i$ is fully faithful (in which case the unit $\mathrm{id}_\mathcal{C} \to i^\ast \circ i_!$ is a natural isomorphism)?
You are asking about the Beck–Chevalley conditions for Kan extensions, which is a fairly non-trivial question. Here is one generic situation where things work. Let $u : \mathcal{A} \to \mathcal{C}$ and $v : \mathcal{B} \to \mathcal{C}$ be functors between small categories, let $(u \downarrow v)$ be the comma category, let $p : (u \downarrow v) \to \mathcal{A}$ and $q : (u \downarrow v) \to \mathcal{B}$ be the projections, and let $\theta : u p \Rightarrow v q$ be the canonical natural transformation. Then, the Beck–Chevalley transformations $$q_! v^* \Rightarrow q_! p^* u^* u_! \Rightarrow q_! q^* v^* u_! \Rightarrow v^* u_!$$ $$u^* v_* \Rightarrow p_* p^* u^* v_* \Rightarrow p_* q^* v^* v_* \Rightarrow p_* q^*$$ are natural isomorphisms. This is Proposition 4.1.19 in my notes.
More generally, there is a notion of exact square, and there is an exact square condition characterising fully faithful functors.