Difference between stationary and non-stationary radiative transfer?

In your case, stationary or non-stationary means time dependence (energy).

Stationary radiative transfer is used as the model in the case of non-interacting particles moving through a medium for example.

The equilibrium distribution of mutually non interacting particles propagating through a scattering medium is described by the stationary radiative transfer equation

https://arxiv.org/abs/1304.6504

Non-stationary radiative transfer is used as the main model in cases where the particles are considered interacting and the energy levels do change with time (time dependence).

The illumination of the dust nebula under the influence of radiation of a new star can be considered as an example of such process. The simplest model of non-stationary radiative transfer is a model based on the consideration of the one-dimensional homogeneous infinite medium with an energy source depending on time.

https://doi.org/10.3103/S106345411701006X