It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of $(\infty,1)$-categories, but I've read that one has a notion of $(\infty,1)$-(co)limit in an $(\infty,1)$-category, for example as explained in this answer. Moreover, I found a paper by D. Gaitsgory which speaks about continuous and cocontinuous (quasi-)functors between dg-categories. Also, cones of closed degree $0$ morphisms in a pretriangulated dg-category are clearly an example of (homotopy) colimit.
So, I believe there is a way to define "dg-(co)limits" in a dg-category. What is a possible pattern? Is there any reference about that?