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In this work, we investigate the magnetic structures of $(Fe_{1-x}Mn_x)_2AlB_2$ solid-solution quaternaries in the $x = 0$ to $1$ range using x-ray and neutron diffraction, magnetization measurements, and mean-field theory calculations. While $Fe_2AlB_2$ and $Mn_2AlB_2$ are known to be ferromagnetic (FM) and antiferromagnetic (AFM), respectively, herein we focused on the magnetic structure of their solid solutions, which is not well understood. The FM ground state of $Fe_2AlB_2$ becomes a canted AFM at $x \approx 0.2$, with a monotonically diminishing FM component until $x \approx 0.5$. The FM transition temperature ($T_C$) decreases linearly with increasing $x$. These changes in magnetic moments and structures are reflected in anomalous expansions of the lattice parameters, indicating a magnetoelastic coupling. Lastly, the magnetocaloric properties of the solid solutions were explored. For $x = 0.2$ the isothermal entropy change is smaller by 30% than it is for $Fe_2AlB_2$, while the relative cooling power is larger by 6%, due to broadening of the temperature range of the transition., Comment: Final version as published in physical review materials