Hartshorne Corollary III.9.4
@poorna In order to apply Prop. 9.3 in the proof of Cor. 9.4 one needs a flat base change
$$u:A \longrightarrow k(y).$$
When $y \in Y$ is not closed then $u$ is not flat: For the proof consider $u$ as the composition $ A \longrightarrow A_{p_y}$ and $A_{p_y} \longrightarrow k(y), y = p_y$. The former morphism is flat being a localisation. Hence we show that the latter is not flat. Flatness of the latter were equivalent to the field $k(y)$ being a free $A_{p_y}$-module of rank 1 (Matsumura, Hideyuki: Commutative ring theory. Theor. 7.10). Apparently, that does not hold because the ideal $p_y \subset A$ is not maximal.
Therefore Hartshorne introduces the intermediate ring $A' := A/p_y$. It is an integral domain. If $Y' := Spec \ A'$ then $y \in Y'$ is the generic point with residue field $k(y) = A'_{(0)}$ a localisation. Hence
$$u':A' \longrightarrow \ k(y)$$
is flat. By definition
$$X_y := X \times_Y Spec \ k(y) = (X \times_{Y} Y') \times_{Y'} Spec \ k(y) = X' \times_{Y'} Spec \ k(y)$$
$$\mathscr F_y := \mathscr F \otimes_{A} k(y) = (\mathscr F \otimes_A A') \otimes_{A'} k(y) = \mathscr F' \otimes_{A'} k(y).$$
Therefore I differ from Hartshorne's notation and propose to set $\mathscr F' := \mathscr F \otimes_A A'$. Was this your point, too?