[1] Processes which are governed by elastic collisions at a microscopic level will appear to be inelastic if a macroscopic observer only has access to a subset of the degrees of freedom.
Deep inelastic scattering of electrons from protons provided the first direct evidence for the existence of quarks.
The blue shift can be observed when internal energy of the matter is transferred to the photon; this process is called anti-Stokes Raman scattering.
Whether elastic or inelastic scatter occurs is dependent on the speed of the neutron, whether fast or thermal, or somewhere in between.
In inelastic scattering, the neutron interacts with the nucleus and the kinetic energy of the system is changed.