This question is related to this question, where I asked about the relation between the derived category of a fiber product $Y \times_Z W$ and the push out of the diagram of derived categories one obtains considering the pullback functors. It was pointed out to me that it was proven by Ben-Zvi, Francis and Nadler, arXiv:0805.0157, that if we consider the derived fiber product, then (under some assumptions) we have an equivalence $QC(Y \times_Z W) \simeq QC(Y) \otimes_{QC(Z)} QC(W)$ where $QC(-)$ denotes the $\infty$ stable category of quasi coherent sheaves and the tensor product is computed in the $\infty$ category of presentable $\infty$ categories with morphisms given by left adjoints. As I am interested in characteristic zero, the formalism of $k$ linear $\infty$ stable categories is equivalent to that of pretriangulated dg categories. I would like to restate the above result in the formalism of pretriangulated dg categories, but I am having some problems doing it. The categories need to be substituted with dg enhancements of the triangulated categories of quasi coherent sheaves and functors of $\infty$ categories need to be substituted with quasi functors, but what is the counterpart of the tensor product? I tried having a look at the paper by Cohn, arXiv:1208.2587, where the equivalence between $k$ linear $\infty$ stable and pretriangulated dg is proven, and as far as I understand one needs to replace one of the two dg categories by a flat dg category. However, I don't understand what being flat for a dg category means, as in the cited paper the definition is given for spectral categories. Would anyone be able to shed some light on this tensor product for me? Thanks.

The derived tensor product of dg-categories was explored by Toën, see his article The homotopy theory of dg-categories and derived Morita theory, in particular, Section 4, where Toën explains how to derive the tensor product of dg-categories.

For a dg-category $\mathcal C$, being flat means that all enriched Hom's are flat (as k-modules). In other words, for every two objects $a,b$, one requires that the functor $\mathcal C(x,y)\otimes-$ preserve quasi-isomorphisms.

See e.g. http://www.mi-ras.ru/~akuznet/dgcat/Keller%20On%20differential%20graded%20categories.pdf

But you are interested in a relative tensor product? In this case I guess that the definition of $\mathcal C$ being flat over $\mathcal A$ (when one has a dg-functor $\mathcal A\to\mathcal C$) is that $\mathcal C$ is flat as an $\mathcal A$-module, that is $\mathcal C\otimes_{\mathcal A}-:\mathcal A-mod\to\mathcal C-mod$ preserves weak-equivalences and colimits.