In her paper "[Mixed perverse sheaves for schemes over number fields][1]" A. Huber defines certain weights for certain categories of $\mathbb{Q}_l$-sheaves over a finite type $\mathbb{Q}$-scheme $S$ using the following argument:

She presents $S$ as a limit of finite type $U_0$-schemes $S_0$, where $U_0$ is an open subscheme of $\mathbb{Z}[1/l]$. Then (as far as I understand) she defines her (derived) category of sheaves over $S$ as a 2-limit of those for all $S_0$. She states that there exist a perverse $t$-structure and a nice notion of weights for this limit category that could be obtained as the limit of the corresponding structures over all $S_0$. So, it seems that the corresponding properties of sheaves and weights over $S_0$ (see Huber's 3.4-3.9) should be true (and the proof is even easier; Huber just relies on the corresponding results of BBD). Is this correct, or is there some matter that simplifies when we pass from $S_0$ to $S$?

  [1]: https://doi.org/10.1023/A:1000273606373 "Compositio Mathematica 108, 107–121 (1997). zbMATH review at https://zbmath.org/?q=an:0882.14006"