Free-Free Continuum Radiation Electron accelerated by an encounter by with a proton but isn't 'captured' by the proton. Synchrotron Continuum Radiation Electrons moving at relativistic velocities across a magnetic field will spiral around the lines of magnetic force. Particle experiences an acceleration. Very different from thermal or free-free. Any mechanism that can produce radiation can also absorb it.
Index of refraction is different at different frequencies.
Different frequencies will travel at different speeds. Some frequencies will arrive earlier or later than others. Makes no difference whether it is the source or observer who is moving. They lose energy in quantized amounts. They can lose energy by emitting light.