  We model the X-ray surface brightness distribution of emission associated
with Fanaroff & Riley type-II radio galaxies. Our approach builds on the RAiSE
dynamical model which describes broadband radio-frequency synchrotron evolution
of jet-inflated lobes in a wide range of environments. The X-ray version of the
model presented here includes: (1) inverse-Compton upscattering of cosmic
microwave background radiation; (2) the dynamics of the shocked gas shell and
associated bremsstrahlung radiation; and (3) emission from the surrounding
ambient medium. We construct X-ray surface brightness maps for a mock catalogue
of extended FR-IIs based on the technical characteristics of the eRosita
telescope. The integrated X-ray luminosity function at low redshifts
($z\leqslant1$) is found to strongly correlate with the density of the ambient
medium in all but the most energetic sources, whilst at high-redshift ($z>1$)
the majority of objects are dominated by inverse-Compton lobe emission due to
the stronger cosmic microwave background radiation. By inspecting our mock
spatial brightness distributions, we conclude that any extended X-ray detection
can be attributed to AGN activity at redshifts $z\geqslant1$. We compare the
expected detection rates of active and remnant high-redshift radio AGNs for
eRosita and LOFAR, and future more sensitive surveys. We find that a factor of
ten more remnants can be detected using X-ray wavelengths over radio
frequencies at $z>2.2$, increasing to a factor of 100 for redshifts $z>3.1$.
