I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one..

Let $X$ be a scheme over a number field $k$. Feel free to add any hypothesis you need or to enlarge the setting (for example if the answer is true for any site over $X$).

Which are the known relations between

-the categories of etale smooth sheaves $\mathbb{Sh}_{\text{smooth}}(\text{Et}/X),\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times\overline{k})$ and $\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times \mathbb{C})$,

-the derived categories $D(\mathbb{Sh}_{\text{smooth}}(\text{Et}/X)),D(\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times\overline{k}))$ and $D(\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times \mathbb{C}))$,

-The etale cohomology groups associated with a complex of the former derived categories.