Axions and axion-like particles (ALPs) are some of the most popular candidates for dark matter, with several viable production scenarios that make different predictions. In the scenario in which the axion is born after inflation, its field develops significant inhomogeneity and evolves in a highly nonlinear fashion. Understanding the eventual abundance and distribution of axionic dark matter in this scenario therefore requires dedicated numerical simulations. So far the community has focused its efforts on simulations of the QCD axion, a model that predicts a specific temperature dependence for the axion mass. Here, we go beyond the QCD axion, and perform a suite of simulations on lattice sizes of 3072 3 , over a range of possible temperature dependencies labelled by a power-law index n ∈ [0, 6]. We study the complex dynamics of the axion field, including the scaling of cosmic strings and domain walls, the spectrum of non-relativistic axions, the lifetime and internal structure of axitons, and the seeds of miniclusters. In particular, we quantify how much the string-wall network contributes to the dark matter abundance as a function of how quickly the axion mass grows. We find that a temperature-independent model produces 25% more dark matter than the standard misalignment calculation. In contrast to this generic ALP, QCD axion models are almost six times less efficient at producing dark matter. Given the flourishing experimental campaign to search for ALPs, these results have potentially wide implications for direct and indirect searches.