Michael Batanin writes (in response to David Carlton):
But now I understand you are asking about weak (or pseudo) colimits. They exist in 1-Cat and 2-Cat and can be expressed in terms of appropriate weighted colimits. I never saw a paper about pseudocolimits in 3-Cat (here we can use Gray-categories instead of general tricatgories). They must be expressed as weighted colimits as well, or a codescent object of a simplicial Gray-category. I'd like to have a reference if such a paper already exists.
Some specific examples of weighted limits in the context of Gray-categories are considered in: Stephen Lack, A coherent approach to pseudomonads, Adv. Math. 152:179-202, 2000. In particular, I show that the (suitably pseudo) Eilenberg-Moore object of a pseudomonad can be computed as a weighted limit. But I don't develop the general theory of pseudocolimits in 3-Cat. Steve Lack.