Divisors whose restriction is big

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?

Consider a product $$X = \mathbb{P}^n\times\mathbb{P}^1$$, with projections $$g:X\rightarrow\mathbb{P}^n$$ and $$f:X\rightarrow\mathbb{P}^1$$.

Set $$H_1:= g^{*}\mathcal{O}_{\mathbb{P}^n}(1)$$ and $$H_2:= f^{*}\mathcal{O}_{\mathbb{P}^1}(1)$$. The effective cone of $$X$$ is closed and generated by $$H_1,H_2$$.

Now, take a divisor $$D = aH_1+bH_2$$ with $$a > 0$$ and $$b < 0$$. Then $$D_{|f^{-1}(p)} = \mathcal{O}_{\mathbb{P}^n}(a)$$, which is ample since $$a > 0$$, for all $$p\in\mathbb{P}^1$$. However, since $$b < 0$$ the divisor $$D$$ is not pseudo-effective.

Take $$Y=\mathbb{P}^1$$ and let $$X=\mathbb{F}_n=\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-n))$$ be the Hirzebruch surface with a section $$C_0$$ such that $$C_0^2=-n$$.

Let $$H$$ be an ample divisor on $$\mathbb{F}_n$$, set $$t:=HC_0$$ and take $$D:=C_0-(t+1)F,$$ where $$F \simeq \mathbb{P}^1$$ is the fibre of $$f \colon \mathbb{F}_n \to \mathbb{P}^1$$.

Finally, take $$\mathcal{L}=\mathcal{O}(1)$$, so that $$A$$ is a point and $$X_A=F$$. The divisor $$D|_{F}$$ has degree $$1$$ on $$F$$, hence $$D|_{F} = \mathcal{O}_F(1)$$ which is ample, in particular big.

On the other hand, we have $$HD=H(C_0-(t+1)F)=t-(t+1)HF <0,$$ and this implies that $$D$$ is not pseudo-effective.