Let $X$ be a quasi-projective scheme over $\mathbb{C}$ and $\frak{X}$ be a (quasi-projective) model over a number field.
Are there relations between the derived categories
$$D(\mathbb{Sh}(\text{Et}/X)),D(\mathbb{Sh}(\text{Et}/\mathfrak{X})) \hspace{2mm} \textit{and} \hspace{2mm} D(\mathbb{Sh}(\text{Et}/\mathfrak{X}\times\overline{\mathbb{Q}}))?$$
If yes, does this generalizes to any derived category of sheaves or is it particular to the etale case?
N.B. I'm adding the hypothesis "quasi-projective" because it is in the setting in which I'm working but feel free to answer in the most generality, or to add hypothesis if necessary.