Thank you very much! This clearifies my ideas. I will think more about examples, and try to post the outcome.

Thank you very much, I will have a look at the references you posted and try to get a better understanding; though my knowledge of mixed Hodge modules is very poor at the moment. The question behind the second paragraph was: is it possible to imitate Deligne's definition for the weights of the Frobenius to get a similar theory of mixed complexes for varieties over $\mathbb{Z}[\frac{1}{p}]$?

Thank you very much Jason, I have been able to understand why this is true. Does this characterization of the relative diagonal as degeneracy locus of a vector bundle admit generalizations to other families (e.g. smooth projective families of curves) ? Of course we will not have these tautological bundles, but I wonder whether it could be possible to construct some other vector bundle which plays a similar role in the description of the relative diagonal.

As I understand you mean that the diagonal coincide with that top Chern class. Is this an obvious fact ?