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Its piezo- and potentially ferroelectric properties make the metastable kappapolymorph 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 theirbandgap. 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 metalexchange catalysis in plasma assisted molecular beam epitaxy (MEXCAT-MBE). Thepolymorph transition from $\kappa$ to $\beta$ is also addressed, highlightingthe 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) attemperatures at least 100 {\deg}C above those achievable with conventionalnon-catalyzed MBE, opening the road for increased crystal quality inheteroepitaxy. The role of the substrate, as well as strain and structuraldefects in the growth of $\kappa$-Ga$_2$O$_3$ is also investigated by growingsimultaneously 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.

Phase-selective growth of $\kappa$- vs $\beta$-Ga$_2$O$_3$ and  (In$_x$Ga$_{1-x}$)$_2$O$_3$ by In-mediated metal exchange catalysis in  plasma-assisted molecular beam epitaxy