In the book Galois Theories by Borceux and Janelidze appears the following lemma.
Lemma 6.2.3. Let $I:\mathsf{Fam}(\mathsf A)\longrightarrow \mathsf{Set}$ be a family fibration and let $D:\mathsf K\longrightarrow \mathsf{Fam}(\mathsf A)$ be a diagram. For every $x=(x_K)_{K\in \mathsf{K}}\in \varprojlim I\circ D$ let $D_x$ be the functor defined by $$(f:K\to K^\prime) \mapsto (D(f)_{x_K}:D(K)_{x_K}\to D(K^\prime)_{x_{K^\prime}}.$$ If each $D_x$ has a limit then $D$ also has a limit, described as one might guess. Moreover, it is preserved by $I$.
After Corollary 6.2.7 appears the following excerpt.
Readers familiar with fibrations of categories will of course recognize that all these observations on limits are special cases of simple and more elegant results (on general fibrations).
What is the general analogue of Lemma 6.2.3 for fibrations? I am not sure where the fibers come in here.
The general analogue of lemma 6.2.3 would be as follow:
The computation goes as follow: take the limiting cone of $pD$ in $\mathscr B$, then pull every $D(i)$ in the fiber above the limit vertex, compute your limit inside this fiber and voilà.
More formally: denote $L$ the limit of $pD$ and $\pi_i : L \to pD(i)$ the canonical projections, and consider the diagram $\tilde D : \mathscr I \to p^{-1}(L)$ given on objects by $i \mapsto \pi_i^\ast(D(i))$ and on morphism by the universal property of the various cartesian morphisms $\pi_i^\ast(D(i)) \to D(i)$. More precisely (draw diagrams to understand this part), for any map $h : i \to j$ in the shape $\mathscr I$, you get a commutative triangle in $\mathscr B$: $$ pD(h)\circ\pi_i = \pi_j$$ As $D(h) \circ (\pi_i^\ast(D(i)) \to D(i))$ is above $pD(h)\circ \pi_i$, we can use the cartesianity of $\pi_j^\ast(D(j)) \to D(j)$ to find an arrow $\pi_i^\ast(D(i)) \to \pi_j^\ast(D(j))$ above $\mathrm{id}_L$ that we define as $\tilde D(h)$. I let you show that it is indeed a functor $\tilde D$ (it uses the unicity in the universal property of cartesian morphisms). Now take the limit $\tilde L$ of $\tilde D$, that is computed in the category $p^{-1}(L)$. I claim it is a limit of $D$.
Why? Well take any cone on $D$ with vertex $X$ and projections $q_i : X \to D(i)$. Then surely it gives a cone over $pD$ with vertex $pX$ and projections $p(q_i)$ in $\mathscr B$. Hence a map $\chi : pX \to L$ using the universal property of the limit $L$. By the universal property of the cartesian morphisms $\pi_i^\ast(D(i)) \to D(i)$, it gives maps $X \to \pi_i^\ast(D(i))$ above $\chi$ for each $i$, and we can show that is in fact a cone over $\tilde D$. But we can't yet conclude using that $\tilde L$ is a limit: indeed, $X$ and the cone we just constructed are not living in the fiber $p^{-1}(L)$ a priori! Hopefully, the cone $(X\to \pi_i^\ast(D(i))_i$ is above the map $\chi$ and so a map of constant cones $X \to \tilde L$ amounts to a map $X \to \chi^\ast(\tilde L)$ in $p^{-1}(pX)$ that is a map of constant cones over the diagram $i \mapsto \chi^\ast\pi_i^\ast(D(i))$. Now we can use the hypothesis that $\chi^\ast$ preserves the diagram of shapes $\mathscr I$ and conclude: $\chi^\ast(\tilde L)$ being the limit of this diagram in $p^{-1}(pX)$, there is such a map $X \to \chi^\ast(\tilde L)$. The unicity of such a map is a consequence of the unicity in the universal property of cartesian morphisms (you should check it!).
What does it have to do with your problem? Well the functor $$I : \mathsf{Fam}(\mathcal A) \to \mathsf{Set}, (a_s)_{s\in S} \mapsto S$$ is a Grothendieck fibration, where for $h : S \to T$, the change of fiber functor is defined as $$ h^\ast : (a_t)_{t\in T} \mapsto (a_h(s))_{s\in S} $$ (or if you prefer the cartesian morphisms above $h$ are those $(f_s)_{s\in S}:(b_s)_{s\in S} \to (a_t)_{t\in T}$ where for each $s\in S$ we have $b_s = a_{h(s)}$ and $f_s = \rm id_{b_s}$).
All you have to do then is to show that those $h^\ast$ preserves small limits. If $\mathcal A$ has small coproducts, then every $h^\ast$ actually has a left adjoint (by Kan extension nonsense) and it is settled. In case $\mathcal A$ does not, I'm inclined to think this is still the case, but have not worked it out. (It should boils down to the fact that Borceux and Janelidze are using implicitely in their proof, that is that limits in the fibers are computed pointwise.)