  With null results in resonance searches at the LHC, the physics potential
focus is now shifting towards the interpretation of non-resonant phenomena. An
example of such shift is the increased popularity of the EFT programme. We can
embark on such programme owing to good integrated luminosity and an excellent
understanding of the detectors, which will allow these searches to become more
intense as the LHC continues. In this paper we provide a framework to perform
this interpretation in terms of a diverse set of scenarios, including {\it 1.)
} generic heavy new physics described at low energies in terms of a derivative
expansion, such as in the EFT approach, {\it 2.)} very light particles with
derivative couplings, such as axions or other light pseudo-Goldstone bosons,
and {\it 3.)} the effect of a quasi-continuum of resonances, which can come
from a number of strongly-coupled theories, extra-dimensional models, clockwork
set-ups and their deconstructed cousins. These scenarios are not equivalent
despite all non-resonant, although the matching among some of them is possible
and we provide it in this paper.
