I am trying to understand the proof of Kashiwara's constructibility theorem for algebraic D-modules, following either the book "D-modules, Perverse sheaves, and representation theory" by Hotta, Takeuchi and Tanisaki, or Bernstein's lecture notes. My question is about theorem 4.7.7 of the D-module book, or theorem C a) on page 39 of Bernstein's note. These theorems say that if $M$ is a holonomic D-module on a smooth algebraic variety $X$ over $\mathbb{C}$, then the de Rham complex $DR_X(M)$ has constructible cohomologies with respect to a stratification by algebraic subsets.

Let us look at the proof of theorem 4.7.7. The proof begins with choosing an open $U\subset X$ on which $M$ is an integrable connection, and let $Z$ be an irreducible component of $X \setminus U$. The aim is to show $DR_X(M)$ has locally constant cohomologies on some dense Zariski open subsets of $Z$. Now one can find subsets $V \subset X$ and $V' \subset \mathbb{A}^n$, and an étale morphism $f: V \to V'$ such that $f^{-1}(V'\cap \mathbb{A}^{n-k}) = Z \cap V = (X \setminus U) \cap V$. Here $n$ and $n-k$ are the dimensions of $X$ and $Z$ respectively.

The book proceeds by saying, $DR_V(M\vert_V)\in D^b_c(V)$ if and only if $f_*(DR_V(M\vert_V))\in D^b_c(V')$ because $f$ is étale, and $f_*(DR_V(M\vert_V)) = DR_{V'} (\int_f^0(M\vert_V))$ by proposition 4.7.5. I don't understand both assertions. Can anyone explain to me why the étale condition implies the first if and only if statement, and why $DR$ commutes with $f_*$ when $f$ is not a priori proper, and why do the authors write $f_*$ and $\int_f^0$ instead of $Rf_*$ and $\int_f$, indicating there is no higher cohomologies?

I understand that the purpose of these steps is to change to the easier situation $X = Z \times \mathbb{P}^k$ and $Z = Z \times pt$, as Bernstein's note says. But Bernstein's note provides even less details than the D-module book I am using, and simply says that by a base change from $\mathbb{A}^k$ to $W$ (where his $\mathbb{A}^k$ is our $\mathbb{A}^{n-k}$ and his $W$ is our $Z$) we can assume $X = W \times \mathbb{P}$ and so on.

Another question of mine is that, How does Bernstein's method work? Does he mean the same thing as authors of the D-module book, or something different?

I am sorry for such a lengthy question, but I can't find a good way to put less words. Thank you very much in ahead of time for reading these paragraphs!