  Multiple relaxation channels often arise in the dynamics of liquids where the
momentum current associated to the particle-conservation law splits into
distinct contributions. Examples are strongly confined liquids for which the
currents in lateral and longitudinal direction to the walls are very different,
or fluids of nonspherical particles with distinct relaxation patterns for
translational and rotational degrees of freedom. Here, we perform an asymptotic
analysis of the slow structural relaxation close to kinetic arrest as described
by mode-coupling theory (MCT) with several relaxation channels. Compared to
standard MCT, the presence of multiple relaxation channels significantly
changes the structure of the underlying equations of motion and leads to
additional, non-trivial terms in the asymptotic solution. We show that the
solution can be rescaled, and thus prove that the well-known $ \beta $-scaling
equation of MCT remains valid even in the presence of multiple relaxation
channels. The asymptotic treatment is validated using a novel schematic model.
We demonstrate that the numerical solution of this schematic model can indeed
be described by the derived asymptotic scaling laws close to kinetic arrest.
Additionally, clear traces of the existence of two distinct decay channels are
found in the low-frequency susceptibility spectrum, suggesting that clear
footprints of the additional relaxation channels can in principle be detected
in simulations or experiments of confined or molecular liquids.
