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The main mechanism responsible for Axion-Like-Particle (ALP) production in the early universe is the so-called misalignment mechanism. Three regimes have been investigated in this context: standard misalignment, large misalignment and kinetic misalignment. The latter applies if the axion inherits a large initial velocity in the early universe, such that the field rolls through many wiggles during its evolution, before it gets trapped in one minimum. This largely opens the region of parameter space for ALP dark matter towards higher values for the axion-photon coupling, which can be probed by the whole set of next decade's upcoming experiments. In fact, almost the entire parameter space in the [mass, decay constant] plane can now accommodate dark matter. In this paper, we show that in kinetic misalignment, the axion field is almost always entirely fragmented, meaning that the energy density of the homogeneous field is redistributed over higher-mode axions. We present a general model-independent analytical description of kinetic fragmentation, including discussion of the modified initial conditions for the mode functions due to the axion's initial velocity, and how they impact the growth of the adiabatic fluctuations. We calculate precisely the parameter regions corresponding respectively to standard misalignment, kinetic misalignment with weak fragmentation, fragmentation after trapping and fragmentation before trapping. While axion fragmentation can impact the precise determination of the relic abundance, another main observational implication is the formation of much denser compact axion halos, that is described in a companion paper. We also point out a new gravitational-wave signature that arises in the large misalignment regime with complete fragmentation and could be seen in measurements of $μ$ distortions in the Cosmic Microwave Background.