  This paper studies the large scale limits of multi-type invariant
distributions and Busemann functions of planar stochastic growth models in the
Kardar-Parisi-Zhang (KPZ) class. We identify a set of sufficient hypotheses for
convergence of multi-type invariant measures of last-passage percolation (LPP)
models to the stationary horizon (SH), which is the unique multi-type
stationary measure of the KPZ fixed point. Our limit theorem utilizes
conditions that are expected to hold broadly in the Kardar-Parisi-Zhang class,
including convergence of the scaled last-passage process to the directed
landscape. We verify these conditions for the six exactly solvable models whose
scaled bulk versions converge to the directed landscape, as shown by Dauvergne
and Vir\'ag. We also present a second, more general, convergence theorem with
potential future applications to polymer models and particle systems. Our paper
is the first to show convergence to the SH without relying on information about
the structure of the multi-type invariant measures of the prelimit models.
These results are consistent with the conjecture that the SH is the universal
scaling limit of multi-type invariant measures in the KPZ class.
