Let $A$ be a regular $\mathbb{C}$-algebra. Let $B_1, B_2$ be quotients of the polynomial ring $B:=A[X_0,X_1,...,X_n]$ for some $n$. Assume that both $B_1$ and $B_2$ are flat over $A$ under the compositions $A \to B \to B_i$, for $i=1,2$. Is the tensor product $B_1 \otimes_{B} B_2$ flat over $A$?

Let $f:X \to Y$ be a flat, proper, surjective morphism between noetherian schemes. Assume $Y$ is irreducible and smooth over $\mathbb{C}$. Suppose that $X$ is the union of two schemes $X_1$ and $X_2$ both flat, proper and surjective over $Y$. Is there any known condition under which the scheme theoretic intersection, $X_1.X_2$ is flat over $Y$ (for example, if the generic fibers of the morphisms $X_1 \to Y$ and $X_2 \to Y$ do not intersect)?