Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor
$$F_U:\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$
I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-colimit of this and obtain a functor
$$\operatorname{2-colim}_U F_U:\textsf{QCoh}(X) \to \textsf{Vect}(K),$$
which should send a quasi-coherent module over $X$ to its stalk on the generic point.