The presence of heavy new physics generally alters the Standard Model (SM) Higgs couplings predictions. In many models, we expect that momentum effects decouple as the new heavy states are integrated out. However, these effects can be important in situations of significant off-shellness at collider experiments. We study the momentum dependence of different beyond the SM scenarios, using form factors to encode the $p^2$ effect on the Higgs couplings. Our focus is on the effects of new strongly interacting sectors, like composite Higgs models with general scalar, vector-like fermions or vectorial resonances. We show a significant enhancement of order $p^2/\Lambda^2$ over the expected $v^2/\Lambda^2$ predictions for the BSM scenarios studied. These effects are competitive with the momentum independent coupling modifications, changing the predictions of the models. Additionally, the use of form factors modifies the shapes of the kinematic distributions, providing new opportunities for LHC signals.