I follow the notations of Milne's lectures notes on etale cohomology, most specifically the section titled "The Gysin map" in chapter 24, p. 145.
Let $k$ be an algebraically closed field, let $Y$ and $X$ be two non singular (separated) complete varieties of respective dimensions $d$ and $a$, and let $\pi: Y\hookrightarrow X$ be a closed immersion. In particular $a \geq d$ and let $c := a - d$ denote the codimension. Let $\Lambda = \mathbb Z / n\mathbb Z$ for some $n$ which is prime with the characteristic of $k$. Since the varieties are complete, I do not bother to distinguish between cohomology with or without compact support.
Functoriality of etale cohomology provides us with restriction maps $\pi^*: \mathrm H^r(X,\Lambda) \to \mathrm H^r(Y,\Lambda)$. Taking duals and applying Poincaré duality as well as some Tate twist, we obtain maps $\pi_*:\mathrm H^r(Y,\Lambda) \to \mathrm H^{r+2c}(X,\Lambda(c))$. These are the Gysin maps.
Let $1_Y$ denote the identity element of $\mathrm H^0(Y,\Lambda) \simeq \Lambda$. The element $\pi_*(1_Y) \in \mathrm H^{2c}(X,\Lambda(c))$ is, by definition, the image of $Y$, seen as a prime cycle on $X$, via the cycle map $\mathrm{cl}_X: \mathrm{CH}^c(X)\to \mathrm H^{2c}(X,\Lambda(c))$.
In Remark 24.2, Milne states the projection formula. As a particular case, for any $x\in \mathrm H^r(X,\Lambda)$, we have the identity $$\pi_*\pi^*(x) = \mathrm{cl}_X(Y) \cup x \in \mathrm H^{r+2c}(X,\Lambda(c)).$$ (Simply take $y = 1_Y \in \mathrm H^0(Y,\Lambda)$ in Milne's notations).
Thus, we have an explicit formula to describe $\pi_*\pi^*: \mathrm H^r(X,\Lambda) \to \mathrm H^{r+2c}(X,\Lambda(c))$.
Is there similarly a formula to compute the reverse composition, ie. $\pi^*\pi_*: \mathrm H^r(Y,\Lambda) \to \mathrm H^{r+2c}(Y,\Lambda(c))$? Would this map be similarly described as a cup product by some canonically defined element of the cohomology of $Y$?