I was considering the following situation. I have a directed/filtering system of categories $C_i$. I understand how take its direct limit (aka colimit) $C$ in the category of categories. My question is:

If $C_i$ is cocomplete is $C$ also cocomplete?

I think in general the answer is no. For example, taking $C_i = C_0 \times C_0 \cdots \times C_0$ for some fixed category $C_0$ and taking $C_i \to C_{i+1}$ to be the obvious inclusion. I don't think the limiting category $C$ has coproducts.

Can I add (non trivial) assumptions so that $C$ suddenly has colimits? (if so, how do I compute them?)

In the situation I am considering I have a bunch more of assumptions, which might be handy. All the categories $C_i$ are Grothendieck abelian. The functors $C_i \to C_j$ are exact, essentially surjective and have right adjoints. The indexing system has an initial element $0$, so any object in $C_i$ comes from $C_0 \to C_i$. Also, all I really want in $C$ are direct limits (ie filtered colimits), not necessarily arbitrary ones.