<p>We investigate a recently proposed non-thermal mechanism for dark matter production, in which a small initial dark matter (<i>χ</i>) number density undergoes exponential growth through scatterings with bath particles (<i>ϕ</i>) in the early universe (<i>χϕ</i> → <i>χχ</i>). The process ends when the scattering rate becomes Boltzmann suppressed. The analysis, in literature, is performed on the simplifying assumption of the dark matter phase space tracing the equilibrium distribution of either standard model or a hidden sector bath. Owing to the non-thermal nature of the production mechanism, this assumption may not necessarily hold. In this work, we test the validity of this assumption by numerically solving the unintegrated Boltzmann equation for the dark matter distribution. Our results, independent of the initial conditions, show that after exponential growth ceases, the dark matter distribution exhibits equilibrium-like behaviour at low comoving momentum, especially for higher couplings. While full kinetic equilibrium-like behaviour is not reached across all momentum modes, the scaled equilibrium approximation provides reasonable estimates for the dark matter abundance. However, for more accurate results in scenarios where dark matter is not in kinetic equilibrium with the thermal bath or does not have sufficiently strong self-interactions with itself that can lead to thermalization, the full unintegrated Boltzmann equation must be solved.</p>