Let $X, Y$ be proper (possibly projective, normal) varieties, and let $\pi: Y \to X$ be a morphism, whose general fiber has dimension $s \geq 0$. Let $\dim X = r$, so that $\dim Y = r + s$. Let $(\dotsm)_Y$ denote the intersection product on $Y$, similar for $X$ and a general fiber $F \subset Y$.
Is there some kind of projection formula, which could go like: If $D_1, \dotsc, D_r$ are divisors on $X$, and $E_1, \dotsc, E_s$ are divisors on $Y$. $$ (\pi^* D_1 \cdot \pi^* D_2 \dotsm \pi^* D_r \cdot E_1 \cdot E_2 \dotsm E_s)_Y = (D_1 \dotsm D_r)_X \cdot (E_1|_F \dotsm E_s|_F)_F.$$
In the case that $s = 0$, this would restrict to the projection formula as found in Debarre's Higher-dimensional algebraic geometry: $$(\pi^*D_1 \dotsm \pi^* D_r)_Y = \deg(\pi) (D_1 \dotsm D_r)_X$$
Does such a theorem, or something similar exists, and what is a good reference for this? I tried to look into Fulton's Intersection theory, but I was a bit intimidated by lots of notation I don't know yet.