Discontinuity of a degenerating escape rate

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 $\mathbb{Q}$.