  We outline a kinetic theory of non-thermal fixed points for the example of a
dilute Bose gas, partially reviewing results obtained earlier, thereby
extending, complementing, generalizing and straightening them out. We study
universal dynamics after a cooling quench, focusing on situations where the
time evolution represents a pure rescaling of spatial correlations, with time
defining the scale parameter. The non-equilibrium initial condition set by the
quench induces a redistribution of particles in momentum space. Depending on
conservation laws, this can take the form of a wave-turbulent flux or of a more
general self-similar evolution, signaling the critically slowed approach to a
non-thermal fixed point. We identify such fixed points using a non-perturbative
kinetic theory of collective scattering between highly occupied long-wavelength
modes. In contrast, a wave-turbulent flux, possible in the perturbative
Boltzmann regime, builds up in a critically accelerated self-similar manner. A
key result is the simple analytical universal scaling form of the
non-perturbative many-body scattering matrix, for which we lay out the concrete
conditions under which it applies. We derive the scaling exponents for the time
evolution as well as for the power-law tail of the momentum distribution
function, for a general dynamical critical exponent $z$ and an anomalous
scaling dimension $\eta$. The approach of the non-thermal fixed point is, in
particular, found to involve a rescaling of momenta in time $t$ by $t^{\beta}$,
with $\beta=1/z$, within our kinetic approach independent of $\eta$. We confirm
our analytical predictions by numerically evaluating the kinetic scattering
integral as well as the non-perturbative many-body coupling function. As a side
result we obtain a possible finite-size interpretation of wave-turbulent
scaling recently measured by Navon et al.
