I am a bit confused on the proof of this lemma. $G$ is a linear algebraic group over an algebraically closed field $k$. A closed subgroup $P$ of $G$ is called parabolic if the quotient variety $G/P$ is complete.
First, where is 5.3.2 (i) used? (This says if $\varphi: X \rightarrow Y$ is an equivariant map of homogeneous $G$-spaces, then $X \times Z \rightarrow Y \times Z$ is an open map for any variety $Z$.) It seems to me that you just take an $E \subseteq G/Q \times X$ closed, then its preimage $A$ in $G \times X$ is a closed set with the property that $(g,x) \in A \Rightarrow (gQ,x) \in A$, and the projection of $A$ onto $X$ is the same as the projection of $E$.
Next, how is the completeness of $P/Q$ being used? What does this have to do with the closed set $\alpha^{-1}A \subseteq P \times G \times X$? Finally, how is the completeness of $G/P$ being used?
I don't think one needs to use 5.3.2(i) here.
Completeness of $P/Q$ is used to to show that the projection of $\alpha^{-1}(A)$ onto $G\times X$ is closed. Completeness of $G/P$ is used after that, in a similar way. This is worked out in nice detail in these notes of Herzig, where it is proposition 117.