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The broad and vague question is in the title. The more precise question is:

Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with limit $\mathcal{C}$. Is $\mathrm{Ind}(\mathcal{C}) \simeq \lim \mathrm{Ind}(\mathcal{C_i})$ ? If not, are there reasonable conditions under which this does work?

[Note: By Theorem 1.1.4.4 in Lurie's "Higher algebra," the limit $\mathcal{C}$ above can be computed in the large world of $\infty$-categories, or in the world of stable $\infty$-categories and exact functors.]

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Let $D$ be the derived category of k-vector spaces, and let $C$ be the part consisting of bounded chain complexes of finite-dimensional vector spaces. $D = \mathit{Ind}(C)$.

Let $P$ be a finite partially ordered set whose nerve is a circle, and consider the constant diagrams $\underline{C}$ and $\underline{D}$ shaped like $P$. The limit of $\underline{D}$ is the category of local systems on a circle. The limit of $\underline{C}$ is the category of local systems on a circle, with bounded finite-dimensional fibers.

I think (local systems on a circle) is not the same as Ind(local systems with bounded finite-dimensional fibers on a circle). The former category contains a compact object whose group of endomorphisms is infinite-dimensional: the local system corresponding to the free rank one k[Z]-module.

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