  In this work we aim to gain qualitative insight on the far-from-equilibrium
behavior of the gluon plasma produced in the early stages of a heavy-ion
collision. It was recently discovered arXiv:1810.10554 that the distribution
functions of quarks and gluons in QCD effective kinetic theory (EKT) exhibit
self-similar "scaling" evolution with time-dependent scaling exponents long
before those exponents reach their pre-hydrodynamic fixed-point values. In this
work we shed light on the origin of this time-dependent scaling phenomenon in
the small-angle approximation to the Boltzmann equation. We first solve the
Boltzmann equation numerically and find that time-dependent scaling is a
feature of this kinetic theory, and that it captures key qualitative features
of the scaling of hard gluons in QCD EKT. We then proceed to study scaling
analytically and semi-analytically in this equation. We find that an
appropriate momentum rescaling allows the scaling distribution to be identified
as the instantaneous ground state of the operator describing the evolution of
the distribution function, and the approach to the scaling function is
described by the decay of the excited states. That is to say, there is a frame
in which the system evolves adiabatically. Furthermore, from the conditions for
adiabaticity we can derive evolution equations for the time-dependent scaling
exponents. In addition to the known free-streaming and BMSS fixed points, we
identify a new "dilute" fixed point when the number density becomes small
before hydrodynamization. Corrections to the fixed point exponents in the
small-angle approximation agree quantitatively with those found previously in
QCD EKT and arise from the evolution of the ratio between hard and soft scales.
