In the hypothesis that the scale of new physics is considerably higher than the energies probed at colliders, we can parametrise modified interactions induced by BSM effects among SM particles in a model-independent framework, the Standard Model Effective Field Theory (SMEFT). The success of indirect searches requires the highest possible level of precision in three complementary components: experimental data, theoretical predictions and a robust framework to globally interpret all subtle deviations from the SM predictions that might arise. In particular, I will discuss whether high-scale new physics effects can be mimicked by low-scale non-perturbative physics, and therefore inadvertently absorbed into the parametrization of the proton structure. In this direction, I will argue about the importance of the simultaneous fit of parton distribution functions (PDFs) and the SMEFT parameter space, quantifying to which extent the BSM effects modify the fitted PDFs and assessing how the resulting bounds on the SMEFT degrees of freedom are modified.