  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 $\mu$ distortions in the Cosmic Microwave Background.
