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.]