Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.

Let $D\subset X$ be a divisor such that $D_{|X_A}$ (the restriction of $D$ to $X_A$) is big for $A\in H^0(Y,\mathcal{L})$ general. Under these conditions, might $D$ be not pseudo-effective?