  We observe a catalytic mechanism during the growth of III-O and IV-O
materials by suboxide molecular-beam epitaxy ($S$-MBE). By supplying the
molecular catalysts In$_2$O and SnO we increase the growth rates of Ga$_2$O$_3$
and In$_2$O$_3$. This catalytic action is explained by a metastable adlayer
$A$, which increases the reaction probability of the reactants Ga$_2$O and
In$_2$O with active atomic oxygen, leading to an increase of the growth rates
of Ga$_2$O$_3$ and In$_2$O$_3$. We derive a model for the growth of binary
III-O and IV-O materials by $S$-MBE and apply these findings to a generalized
catalytic description for metal-oxide catalyzed epitaxy (MOCATAXY), applicable
to elemental and molecular catalysts. We derive a mathematical description of
$S$-MBE and MOCATAXY providing a computational framework to set growth
parameters in previously inaccessible kinetic and thermodynamic growth regimes
when using the aforementioned catalysis. Our results indicate MOCATAXY takes
place with a suboxide catalyst rather than with an elemental catalyst. As a
result of the growth regimes achieved, we demonstrate a Ga$_2$O$_3$/Al$_2$O$_3$
heterostructure with unrivaled crystalline quality, paving the way to the
preparation of oxide device structures with unprecedented perfection.
