Let $k^{a}$ be an algebraic closure of $k$ and let $V^{\prime}$ be an irreducible component of (the reduction of) $V^{a}:=V\otimes k^{a}$. There exists a component $W^{\prime}$ of $W^{a}$ dominating $V^{\prime}$. Upon replacing $W$ and $V$ by $W^{\prime}$ and $V^{\prime}$ we are reduced to the case where $k$ is algebraically closed. Upon taking successive hyperplane sections of $W$ not containing any component of $F$ and choosing an irreducible component dominating $V$ we may then assume that $\mu$ is generically finite. In that case we have

$$(\deg\mu)\left(\alpha\cdot\beta^{\dim V-1}\right)_{V}=\left(\mu^{*}\alpha\cdot\mu^{*}\beta^{\dim V-1}\right)_{W}$$

and the result is then clear since $\mu^{*}\beta$ is nef. ∎