I'm not familiar with derived tensor product, hence it may be a stupid question.
Let $f: X\to Y$ be a separated morphism of finite type between schemes. Let $A$ be a torsion ring, and $D^-(X,A)$ (resp. $D^-(k,A)$) be the derived category of bounded above complexes of étale sheaves over $X_{ét}$ (resp. $Y_{ét}$). Suppose that $K\in D^-(X,A)$, $L\in D^-(Y,A)$, then we have projection formula $$L\otimes_A^Lf_!K\cong Rf_!(f^*L\otimes_A^LK)$$ (Ref. Lei FU Etale Cohomology Theory)
Some references apply it (or some $\ell$-adic version of it) to obtain the following result:
Let $k$ be a finite field of characteristic $p$ with an algebraic closure $\overline{k}$, $f:X\to \mathrm{Spec}\overline{k}$ is a separated morphism of finite type. $\ell\neq p$ is a prime. $\mathcal{F}$ is a lisse $\overline{\mathbb{Q}}_\ell$-sheaf on $X$. For each integer $n$, we associate a rank one lisse $\overline{\mathbb{Q}}_\ell$-sheaf $\overline{\mathbb{Q}}_\ell(n)$ on $\mathrm{Spec}k$ such that, taking stalks, the induced character of $\pi_1^{ét}(k)=\mathrm{Gal}(\overline{k}/k)$ takes the value $\mathrm{Card}(k)^{-n}$ (maybe it is some kind of Tate twist?). Its pullback on $\mathrm{Spec}{\overline{k}}$ is still denoted by $\overline{\mathbb{Q}}_\ell(n)$. $\mathcal{F}(n):=\mathcal{F}\otimes f^*\overline{\mathbb{Q}}_\ell(n)$, then for any integer $n$, we have $$H^i_c(X,\mathcal{F}(n))\cong H_c^i(X,\mathcal{F})(n)$$ (e.g. Nicholas M. Katz, Peter Sarnak Random Matrices, Frobenius Eigenvalues, and Monodromy 9.0.13)
However, I found no details of the computation and I cannot give the details. Could you kindly fill in the details? Or is there any detailed reference?