Star formation in our Galaxy occurs in molecular clouds that are self-gravitating, highly turbulent, and magnetized. We study the conditions under which cloud cores inherit large-scale magnetic field morphologies and how the field is governed by cloud turbulence. We present four moving-mesh simulations of supersonic, turbulent, isothermal, self-gravitating gas with a range of magnetic mean-field strengths characterized by the Alfvenic Mach number $mathcal{M}_{{rm A}, 0}$, resolving pre-stellar core formation from parsec to a few AU scales. In our simulations with the turbulent kinetic energy density dominating over magnetic pressure ($mathcal{M}_{{rm A}, 0}>1$), we find that the collapse is approximately isotropic with $Bproptorho^{2/3}$, core properties are similar regardless of initial mean-field strength, and the field direction on $100$ AU scales is uncorrelated with the mean field. However, in the case of a dominant large-scale magnetic field ($mathcal{M}_{{rm A}, 0}=0.35$), the collapse is anisotropic with $Bproptorho^{1/2}$. This transition at $mathcal{M}_{{rm A}, 0}sim1$ is not expected to be sharp, but clearly signifies two different paths for magnetic field evolution in star formation. Based on observations of different star forming regions, we conclude that star formation in the interstellar medium may occur in both regimes. Magnetic field correlation with the mean-field extends to smaller scales as $mathcal{M}_{{rm A}, 0}$ decreases, making future ALMA observations useful for constraining $mathcal{M}_{{rm A}, 0}$ of the interstellar medium.
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