For a graph invariant π, the Contraction(π) problem consists of, given a graph G and positive integers k, d, deciding whether one can contract k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] studied the case where π is the size of a minimum dominating set. We focus on graph invariants defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection H according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in H, in particular implying that Contraction(π) is co-NP-hard for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.