First of all, let me see if I got the 1-categorical version right:

 - Let $\mathcal  F:C\to Cat $ be a
   (pseudo-) functor. The 2-colimit
   $\mathrm{colim}_C\mathcal F$ is then
   given by the Grothendieck
   construction $\int_C \mathcal F$ and
   the 2-limit is given by the category
   of Cartesian sections of the
   fibration $\int_C \mathcal F\to  C$, right?

Can this be transported to the setting of dg$_k$-categories? So: 

 1. Is there a notion of fibration of dg
    categories? I would imagine them to
    be algebras for a dg-monad
    $(\mathrm{id}_C,-)$ arising from
    forming dg-comma categories with the
    identity-span on $C$.
 2. What about a
    Grothendieck construction for
    functors $\mathcal F: C\to \mathrm{dgCat}$ from
    a category $C$ to dg-Categories?
 3. Cartesian sections should then be
    defined as algebra-morphisms from
    the identity on $C$ to $\int_C
    \mathcal F$.

I think there are some problems with what i just said: What are dg-comma categories? What are the right functors $C\to\mathrm{dgCat}$? (I guess one should build a dg-category $C'$ out of $C$ by taking the free $k$-category and then consider it as a dg-category, concentrated in degree 0, and then consider $\mathrm{dgCat}(C',\mathrm{dgCat})$) Same goes for the definition of an algebra morphism: What are the coherences to consider?

So I guess the core question is:

 - What is the right notion of limit for dg categories? (And why?)