  In this work we discuss the emergence of approximate causality in a general
setup from waveguide QED -i.e. a one-dimensional propagating field interacting
with a scatterer. We prove that this emergent causality translates into a
structure for the N-photon scattering matrix. Our work builds on the derivation
of a Lieb-Robinson-type bound for continuous models and for all coupling
strengths, as well as on several intermediate results, of which we highlight
(i) the asymptotic independence of space-like separated wave packets, (ii) the
proper definition of input and output scattering states, and (iii) the
characterization of the ground state and correlations in the model. We
illustrate our formal results by analyzing the two-photon scattering from a
quantum impurity in the ultrastrong coupling regime, verifying the cluster
decomposition and ground-state nature. Besides, we generalize the cluster
decomposition if inelastic or Raman scattering occurs, finding the structure of
the S-matrix in momentum space for linear dispersion relations. In this case,
we compute the decay of the fluorescence (photon-photon correlations) caused by
this S-matrix.
