A high-throughput investigation of local epitaxy (called combinatorial substrate epitaxy) was carried out on Ca$_2$MnO$_4$ Ruddlesden-Popper thin films of six thicknesses (from 20 to 400 nm), all deposited on isostructural polycrystalline Sr$_2$TiO$_4$ substrates. Electron backscatter diffraction revealed grain-over-grain local epitaxial growth for all films, resulting in a single orientation relationship ($OR$) for each substrate-film grain pair. Two preferred epitaxial $ORs$ accounted for more than 90 % of all ORs on 300 different microcrystals, based on analyzing 50 grain pairs for each thickness. The unit cell over unit cell $OR$ ([100][001]$_{film}$ $parallel$ [100][001]$_{substrate}$, or $OR1$) accounted for approximately 30 % of each film. The $OR$ that accounted for 60 % of each film ([100][001]$_{film}$ $parallel$ [100][010]$_{substrate}$, or $OR2$) corresponds to a rotation from $OR1$ by 90$^{circ}$ about the a-axis. $OR2$ is strongly favored for substrate orientations in the center of the stereographic triangle, and $OR1$ is observed for orientations very close to (001) or to those near the edge connecting (100) and (110). While $OR1$ should be lower in energy, the majority observation of $OR2$ implies kinetic hindrances decrease the frequency of $OR1$. Persistent grain over grain growth and the absence of variations of the $OR$ frequencies with thickness implies that the growth competition is finished within the first few si{ anometer}, and local epitaxy persists thereafter during growth.
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