Its large intrinsic polarization makes the metastable κ-Ga2O3 polymorph appealing for multiple applications, and the In-incorporation into both κ and β-Ga2O3 allows us to engineer their bandgap on the low-end side. In this work, we provide practical guidelines to grow thin films of single phase κ-, β-Ga2O3 as well as their (InxGa1−x)2O3 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 role of substrate temperature, oxidizing power, growth rate, and choice of substrate on phase formation and In-incorporation is investigated. As a result, the κ phase can be stabilized in a narrow deposition window irrespective of the choice of substrate [(i) α-Al2O3 (0001), (ii) 20 nm of (2̄01) β-Ga2O3 on α-Al2O3 (0001), and (iii) (2̄01) β-Ga2O3 single crystal]. Low growth rates/metal fluxes as well as growth temperatures above 700 °C tend to stabilize the β-phase independently. Lower growth temperatures and/or O-richer deposition atmospheres allow to increase the In-incorporation in both polymorphs. Finally, we also demonstrate the possibility to grow (2̄01) β-Ga2O3 on top of α-Al2O3 (0001) at temperatures at least 100 °C above those achievable with conventional non-catalyzed MBE, opening the road for better crystal quality in heteroepitaxy.