Let $\operatorname{ProjSmooth}_k$ be the category of smooth projective varieties over a field $k$, and consider the $\ell$ cohomology theory $H^*(-)$ ($l \not = \operatorname{char} k$), how to define reasonable trace maps $\operatorname{Tr}:H^*(X) \rightarrow H^*(Y)$ for any finite flat morphism $f: X \rightarrow Y$ in $\operatorname{ProjSmooth}_k$?

In particulal, trace maps shall be compatible with base change and $ \operatorname{Tr} \circ f^* = \deg f$. In the case $f$ is etale, one can define it using adjunction between $f^!$ and $f_!$ . But what if we only assume the flatness condition?

Motivation: We can define trace maps of finite flat morphisms for coherent cohomology or algebraic de Rham cohomology.