I can only offer a partial answer:

First, have a look at an explicit description of [limits of categories][1].

Namely, strong (or pseudo-) limits can be modelled as the category of cartesian sections into the Grothendieck construction. Denote this category $\Gamma_{\mathrm{Gr}(F)}$.

[Part I] Existence of $\kappa$-directed limits:

Now; Let $F:X\to\mathrm{Cat}$ be a strong functor such that all the functors $F(f):F(x)\to F(y)$ involved are $\kappa$-accessible. Then $\kappa$-filtered colimits can be computed pointwise.

More generally, consider a functor $\gamma:I\to \Gamma_{\mathrm{Gr}(F)}$ and assume all the functors $F(f)$ to preserve $I$-indexed colimits. Then for every $x\in X$ we get a functor $\gamma_{(-)}(x):I\to F(x)$. Then take $\gamma^*:X\to\Gamma_{\mathrm{Gr}(F)}$ to be the functor given on objects by
$$x\mapsto \mathrm{colim}_i\,\gamma_i(x).$$
The morphism part of this functor is defined using the universal properties of the colimits involved. If i'm not mistaken, $\gamma^*$ is a colimit of $\gamma$ and again a cartesian section.

As a special case take $I$ to be any $\kappa$-directed poset.

[Part II] Generated by $\kappa$-compact objects?

I'm not so sure about the second part of the definition of [$\kappa$-accessible category][2].

My idea would be to consider the case where all the $F(f)$ preserve $\kappa$-compact objects. This is certainly the case in the article you linked to as Paré and Rosický consider fully faithful functors.

The **problem** now is that i don't see how one could extend a $\kappa$-compact object in one fiber $F(x)$ to a global cartesian section: Even though the stalks/fibers might be generated by $\kappa$-compact objects i don't know how to even of the existence of such objects in the category of (cartesian) sections.

But perhaps one of the other posters can say something in this direction.

  [1]: http://mathoverflow.net/questions/137689/explicit-description-of-the-oplax-limit-of-a-functor-to-cat
  [2]: http://ncatlab.org/nlab/show/accessible+category