  Its piezo- and potentially ferroelectric properties make the metastable kappa
polymorph of Ga$_2$O$_3$ an interesting material for multiple applications,
while In-incorporation into any polymorphs of Ga$_2$O$_3$ allows to lower their
bandgap. In this work, we provide a guideline to achieve single phase
$\kappa$-, $\beta$-Ga$_2$O$_3$ as well as their (In$_x$Ga$_{1-x}$)$_2$O$_3$
alloys up to x = 0.14 and x = 0.17 respectively, using In-mediated metal
exchange catalysis in plasma assisted molecular beam epitaxy (MEXCAT-MBE). The
polymorph transition from $\kappa$ to $\beta$ is also addressed, highlighting
the fundamental role played by the thermal stability of the
$\kappa$-Ga$_2$O$_3$. Additionally, we also demonstrate the possibility to grow
($\bar{2}$01) $\beta$-Ga$_2$O$_3$ on top of $\alpha$-Al$_2$O$_3$ (0001) at
temperatures at least 100 {\deg}C above those achievable with conventional
non-catalyzed MBE, opening the road for increased crystal quality in
heteroepitaxy. The role of the substrate, as well as strain and structural
defects in the growth of $\kappa$-Ga$_2$O$_3$ is also investigated by growing
simultaneously on three different materials: (i) $\alpha$-Al$_2$O$_3$ (0001),
(ii) 20 nm of ($\bar{2}$01) $\beta$-Ga$_2$O$_3$ on $\alpha$-Al$_2$O$_3$ (0001)
and (iii) ($\bar{2}$01) $\beta$-Ga$_2$O$_3$ single crystal.
