  Motivated by RIXS experiments on a wide range of complex heavy oxides,
including rhenates, osmates, and iridates, we discuss the theory of RIXS for
site-localized $t_{2g}$ orbital systems with strong spin-orbit coupling. For
such systems, we present exact diagonalization results for the spectrum at
different electron fillings, showing that it accesses "single-particle" and
"multi-particle" excitations. This leads to a simple picture for the energies
and intensities of the RIXS spectra in Mott insulators such as double
perovskites which feature highly localized electrons, and yields estimates of
the spin-orbit coupling and Hund's coupling in correlated $5d$ oxides. We
present new higher resolution RIXS data at the Re-L$_3$ edge in Ba$_2$YReO$_6$
which finds a previously unresolved peak splitting, providing further
confirmation of our theoretical predictions. Using ab initio electronic
structure calculations on Ba$_2$${\cal M}$ReO$_6$ (with ${\cal M}$=Re, Os, Ir)
we show that while the atomic limit yields a reasonable effective Hamiltonian
description of the experimental observations, effects such as $t_{2g}$-$e_g$
interactions and hybridization with oxygen are important. Our ab initio
estimate for the strength of the intersite exchange coupling shows that,
compared to the osmates, the exchange is one or two orders of magnitude weaker
in the rhenates and iridates, which may partly explain the suppression of
long-range magnetic order in the latter compounds. As a way to interpolate
between the site-localized picture and our electronic structure band
calculations, we discuss the spin-orbital levels of the ${\cal M}$O$_6$
cluster. This suggests a possible role for non-dispersive intra-cluster
excitons in Ba$_2$YIrO$_6$ which may lead to a weak breakdown of the atomic
$J_{\rm eff}=0$ picture and to small magnetic moments.
