Let $C$ be a curve in a nice smooth surface $S$. Thus $C$ is Cartier and the normal bundle is $N=O(C)_C$, so that deformations of $C$ should intersect $C$ locally in a divisor that is equivalent to $N$ in $Pic(C)$. In the simple examples I know, along ALL the deformation this property holds (i.e any $C'$ in the deformation intersects $C$ in a divisor that is like $N$). I would like to see an example where this holds only locally; so I want a family of curves in a surface $S$, at time $0$ it's $C$, and for other times, $C_t$ intersect $C$ transversely in a nonconstant class group divisor (nonconstant along $t$).
Here are my failed attempts at examples in surfaces:
If $S = \mathbb{P}^2$, then any deformation will have fibers curve that are the same in $Pic(S)$ and so their intersections with $C$ will be the same. (This is because the degree of the curves along the deformation is constant by flatness, so any time $C'$ is transverse to $C$ the intersection will be the same in the class group).
To avoid the problem of 1, we might as well start by searching for a deformation of line bundles that aren't all isomorphic. This can be done via take an elliptic curve $E$, and the family over $E$ of its points (aka line bundles). If we now try to turn this into example we can try taking $S = E x D$ for some $D$, but then the fibers have trivial normal bundles and I don't see how to build a family that has any intersection with some fiber.