The axion field, the angular direction of the complex scalar field associated with the spontaneous symmetry breaking of the Peccei–Quinn (PQ) symmetry, could have originated with initial non-zero velocity. The presence of a non-zero angular velocity resulting from additional terms in the potential that explicitly break the PQ symmetry has important phenomenological consequences such as a modification of the axion mass with respect to the conventional PQ framework or an explanation for the observed matter-antimatter asymmetry. We elaborate further on the consequences of the “kinetic misalignment” mechanism, assuming that axions form the entirety of the dark matter abundance. The kinetic misalignment mechanism possesses a weak limit in which the axion field starts to oscillate at the same temperature as in the conventional PQ framework, and a strong limit corresponding to large initial velocities which effectively delay the onset of oscillations. Following a UV-agnostic approach, we show how this scenario impacts the formation of axion miniclusters, and we sketch the details of these substructures along with potential detecting signatures.