We consider the momentum-dependent effects in Higgs couplings generated by physics beyond the standard model. We take a model-dependent approach, in which we can fully compute the non-local effects from physics not directly reachable by the LHC energy. We consider several scenarios, including composite Higgs models, additional scalars, and the continuum contributions of a quasi-conformal sector, as examples. For each specific model, we are able to obtain the form factor, with which it is then possible to fully simulate the effects in kinematics distributions. The momentum-dependent effects appear as a consequence of off-shellness in the process. We show how the sensitivity of different channels to the various models depends on how the flow of off-shellness appears in the Higgs couplings.