We study constraints on new physics from Higgs production at the LHC in the context of an effective field theory (EFT), focusing on Higgs searches in HV (V = W, Z) associated production which are particularly sensitive to the high-energy behavior of certain dimension-six operators. We show that analyses of these searches are generally dominated by a kinematic region where the generic EFT expansion breaks down, and we establish under which conditions they can nevertheless be meaningful. For example, constraints from these searches on the Wilson coefficients of operators whose effects grow with energy can be established in scenarios where a particular combination of fermions and the Higgs are composite and strongly coupled: then, bounds from Higgs physics at high energy are complementary to LEP1 and competitive with LEP2.