Let $\pi\colon Z \to \Delta$ be a smooth family of complex (projective) varieties, over a small disk in $\mathbb{C}$ such that $\pi^{-1}(0)$ is the only (normal-crossing) singular fiber, $\pi^{-1}(0)= X\cup_D Y$.

I have some questions (may be equal) about line bundle on the family vs individual fiber.

Under what condition (on $X$, $Y$, $D$), it is true that we extend a line bundle on the singular fiber to a line bundle on the whole family?

Under what condition (on $X$, $Y$, $D$), it is true that there is an isomorphism between the space of line bundles (Picard group) on the singular fiber and the space of line bundles on the smooth fiber?

I general I would like to know about the comparison between the Picard group of $Z$ and its fibers.

For $t\neq 0$, I expect Pic($Z_t$) to be bigger than Pic($Z_0$), can you provide an example of that?