  We look at degenerating meromorphic families of rational maps on
$\mathbb{P}^1$ -- holomorphically parameterized by a punctured disk -- and we
provide examples where the bifurcation current fails to have a bounded
potential in a neighborhood of the puncture. This is in contrast to the recent
result of Favre-Gauthier that we always have continuity across the puncture for
families of polynomials; and it provides a counterexample to a conjecture posed
by Favre in 2016. We explain why our construction fails for polynomial families
and for families of rational maps defined over finite extensions of the
rationals $\mathbb{Q}$.
