By linearity we reduce to the case where $\alpha = [V]$ and $\beta = [W]$ for some closed subvariety $V \subset X$ and $W \subset Y$ of dimension $r$ and $s$. Then $f^{-1}(W)$ has pure dimension $s + e$. We assume the cycles $[V]$ and $f^*[W]$ intersect properly. We will use without further mention the fact that $V \cap f^{-1}(W) \to f(V) \cap W$ is surjective. \medskip\noindent Let $a$ be the dimension of the generic fibre of $V \to f(V)$. If $a > 0$, then $f_*[V] = 0$. In particular $f_*\alpha$ and $\beta$ intersect properly. To finish this case we have to show that $f_*([V] \cdot f^*[W]) = 0$. However, since every fibre of $V \to f(V)$ has dimension $\geq a$ (see Morphisms, Lemma \ref{morphisms-lemma-openness-bounded-dimension-fibres}) we conclude that every irreducible component $Z$ of $V \cap f^{-1}(W)$ has fibres of dimension $\geq a$ over $f(Z)$. This certainly implies what we want. \medskip\noindent Assume that $V \to f(V)$ is generically finite. Let $Z \subset f(V) \cap W$ be an irreducible component. Let $Z_i \subset V \cap f^{-1}(W)$, $i = 1, \ldots, t$ be the irreducible components of $V \cap f^{-1}(W)$ dominating $Z$. By assumption each $Z_i$ has dimension $r + s + e - \dim(X) = r + s - \dim(Y)$. Hence $\dim(Z) \leq r + s - \dim(Y)$. Thus we see that $f(V)$ and $W$ intersect properly, $\dim(Z) = r + s - \dim(Y)$, and each $Z_i \to Z$ is generically finite. In particular, it follows that $V \to f(V)$ has finite fibre over the generic point $\xi$ of $Z$. Thus $V \to Y$ is finite in an open neighbourhood of $\xi$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}. Using a very general projection formula for derived tensor products, we get $$ Rf_*(\mathcal{O}_V \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{O}_W) = Rf_*\mathcal{O}_V \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{O}_W $$ see Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change}. Since $f$ is flat, we see that $Lf^*\mathcal{O}_W = f^*\mathcal{O}_W$. Since $f|_V$ is finite in an open neighbourhood of $\xi$ we have $$ (Rf_*\mathcal{F})_\xi = (f_*\mathcal{F})_\xi $$ for any coherent sheaf on $X$ whose support is contained in $V$ (see Cohomology of Schemes, Lemma \ref{coherent-lemma-higher-direct-images-zero-finite-fibre}). Thus we conclude that \begin{equation} \label{equation-stalks} \left( f_*\text{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_V, f^*\mathcal{O}_W) \right)_\xi = \left(\text{Tor}_i^{\mathcal{O}_Y}(f_*\mathcal{O}_V, \mathcal{O}_W)\right)_\xi \end{equation} for all $i$. Since $f^*[W] = [f^*\mathcal{O}_W]_{s + e}$ by Lemma \ref{lemma-pullback} we have $$ [V] \cdot f^*[W] = \sum (-1)^i [\text{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_V, f^*\mathcal{O}_W)]_{r + s - \dim(Y)} $$ by Lemma \ref{lemma-tor-sheaf}. Applying Lemma \ref{lemma-push-coherent} we find $$ f_*([V] \cdot f^*[W]) = \sum (-1)^i [f_*\text{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_V, f^*\mathcal{O}_W)]_{r + s - \dim(Y)} $$ Since $f_*[V] = [f_*\mathcal{O}_V]_r$ by Lemma \ref{lemma-push-coherent} we have $$ [f_*V] \cdot [W] = \sum (-1)^i [\text{Tor}_i^{\mathcal{O}_X}(f_*\mathcal{O}_V, \mathcal{O}_W)]_{r + s - \dim(Y)} $$ again by Lemma \ref{lemma-tor-sheaf}. Comparing the formula for $f_*([V] \cdot f^*[W])$ with the formula for $f_*[V] \cdot [W]$ and looking at the coefficient of $Z$ by taking lengths of stalks at $\xi$, we see that (\ref{equation-stalks}) finishes the proof.