  We describe the effect of ramified morphisms on Harbourne constants of
reduced effective divisors. With this goal, we introduce the pullback of a
weighted cluster of infinitely near points under a dominant morphism between
surfaces, and describe some of its basic properties. As an application, we
describe configurations of curves with transversal intersections and $H$-index
arbitarily close to $-25/7\simeq -3.571$, smaller than any previously known
result.
