  Some beyond $\Lambda$CDM cosmological models have dark-sector energy
densities that suffer phase transitions. Fluctuations entering the horizon
during such a transition can receive enhancements that ultimately show up as a
distinctive bump in the power spectrum relative to a model with no phase
transition. In this work, we study the non-linear evolution of such signatures
in the matter power spectrum and correlation function using N-body simulations,
perturbation theory and HMcode - a halo-model based method. We focus on
modelling the response, computed as the ratio of statistics between a model
containing a bump and one without it, rather than in the statistics themselves.
Instead of working with a specific theoretical model, we inject a parametric
family of Gaussian bumps into otherwise standard $\Lambda$CDM spectra. We find
that even when the primordial bump is located at linear scales, non-linearities
tend to produce a second bump at smaller scales. This effect is understood
within the halo model due to a more efficient halo formation. In redshift space
these nonlinear signatures are partially erased because of the damping along
the line-of-sight direction produced by non-coherent motions of particles at
small scales. In configuration space, the bump modulates the correlation
function reflecting as oscillations in the response, as it is clear in linear
Eulerian theory; however, they become damped because large scale coherent flows
have some tendency to occupy regions more depleted of particles. This mechanism
is explained within Lagrangian Perturbation Theory and well captured by our
simulations.
