  Over the past years, there has been a sustained effort to systematically
enhance our understanding of medium-induced emissions occurring in the
quark-gluon plasma, driven by the ultimate goal of advancing our comprehension
of jet quenching phenomena. To ensure meaningful comparisons between these new
calculations and experimental data, it becomes crucial to model the interplay
between the radiation process and the evolution of the medium parameters,
typically described by a hydrodynamical simulation. This step presents
particular challenges when dealing with calculations involving the resummation
of multiple scatterings, which have been shown to be necessary for achieving an
accurate description of the in-medium emission process. In this paper, we
extend our numerical calculations of the fully-resummed gluon spectrum to
account for longitudinally expanding media. This new implementation allows us
to quantitatively assess the accuracy of previously proposed scaling laws that
establish a correspondence between an expanding medium and a "static
equivalent". Additionally, we show that such scaling laws yield significantly
improved results when the static reference case is replaced by an expanding
medium with the temperature following a simple power-law decay. Such
correspondence will enable the application of numerical calculations of
medium-induced energy loss in realistic evolving media for a broader range of
phenomenological studies.
