The influence of In on the growth of ɛ-Ga2O3 by plasma-assisted molecular beam epitaxy is investigated. We demonstrate alloying of ɛ-Ga2O3 with In and describe its incorporation limits and catalytic effect on the growth kinetics. A special focus lies on the metal-rich growth regime, where we show that In acts as a surfactant for the ɛ-Ga2O3 growth. Both the In-incorporation and the ɛ-(In,Ga)2O3 growth rate follow a non-monotonous trend with increasing In-supply. Whereas both entities show an increase for low In-fluxes, they decrease again for very high In-fluxes, combined with the formation of an atomically smooth surface for layers with thicknesses of several hundred nanometers in this regime. Based on these results, the influence of the growth temperature and the amount of Sn, supplied to initiate the ɛ-Ga2O3 phase formation, is discussed, revealing their impact on the surfactant ability and incorporation of In. Using In as a surfactant, we demonstrate the pseudomorphic growth of ɛ-(In,Ga)2O3/ɛ-(Al,In,Ga)2O3 heterostructures with sharp interfaces and surfaces.