  We consider the dynamics of a separable Continuous Time Random Walk (CTRW)
when the random walker is biased by a velocity field in a uniformly growing
domain. Concrete examples for such domains include growing biological cells or
lipid vesicles, biofilms and tissues, but also macroscopic systems such as
expanding aquifers during rainy periods, or the expanding universe. The
considered CTRW can be subdiffusive, normal diffusive or superdiffusive,
including the particular case of a L\'evy flight. We first consider the case of
zero velocity field. In the subdiffusive case, we reveal an interesting time
dependence of the kurtosis of the particle probability density function. In
particular, for a suitable parameter choice, we find that the propagator, which
is fat tailed at short times, may cross over to a Gaussian-like propagator. We
subsequently incorporate the effect of the velocity field and derive a
bi-fractional diffusion-advection equation encoding the time evolution of the
particle distribution. We apply this equation to study the mixing kinetics of
two diffusing pulses, whose peaks move towards each other under the action of
velocity fields acting in opposite directions. This deterministic motion of the
peaks, together with the diffusive spreading of each pulse, tends to increase
particle mixing, thereby counteracting the peak separation induced by the
domain growth. As a result of this competition, different regimes of mixing
arise. In the case of L\'evy flights, apart from the non-mixing regime, one has
two different mixing regimes in the long-time limit, depending on the exact
parameter choice: In one of these regimes, mixing is mainly driven by diffusive
spreading, while in the other mixing is controlled by the velocity fields
acting on each pulse. Possible implications for encounter-controlled reactions
in real systems are discussed.
