I don't know if this qualifies, because like Daniel Litt, I'm not sure I understand the question. Nonetheless...

Frequently in algebraic geometry to show $\chi_0$ has some open niceness property, e.g. being Cohen-Macaulay, we make a family $\chi$ all of whose fibers are isomorphic to $\chi_0$ (perhaps by being translates under a group action), except for one "bad" one, $\chi_\infty$. Then show that $\chi_\infty$ is good, thereby indirectly inferring that $\chi_0$ must be good.

(Niceness properties seems to almost exactly be the ones that are open in families. A counterexample is given by "normal crossings divisor", a deformation of which may not have normal crossings.)