An object in a category is called compact if the functor corepresented by it commutes with filtered colimits. Now what are the compact objects in the category of $n$-categories?
It was suggested in the comments, that it‘s the $n$-categories that are somehow finitely presented. I would be very happy about an exact formulation and then a proof, a reference or a counterexample to this claim.
Maybe it helps though, to consider the case $n = 1$ first. One can embed $1$-categories into simplicial sets (a similar thing is possible with $n$-categories which can be embedded into $\Theta_{n}$-sets). In simplicial sets, the compact objects are finite colimits of representables as per this question. But the inclusion does not commute with colimits, so I am not sure whether this will be useful in the end.