  Expanding media are typical in many different fields, e.g. in Biology and
Cosmology. In general, a medium expansion (contraction) brings about dramatic
changes in the behavior of diffusive transport properties. Here, we focus on
such effects when the diffusion process is described by the Continuous Time
Random Walk (CTRW) model. For the case where the jump length and the waiting
time probability density functions (pdfs) are long-tailed, we derive a general
bifractional diffusion equation which reduces to a normal diffusion equation in
the appropriate limit. We then study some particular cases of interest,
including L\'evy flights and subdiffusive CTRWs. In the former case, we find an
analytical exact solution for the Green's function (propagator). When the
expansion is sufficiently fast, the contribution of the diffusive transport
becomes irrelevant at long times and the propagator tends to a stationary
profile in the comoving reference frame. In contrast, for a contracting medium
a competition between the spreading effect of diffusion and the concentrating
effect of contraction arises. For a subdiffusive CTRW in an exponentially
contracting medium, the latter effect prevails for sufficiently long times, and
all the particles are eventually localized at a single point in physical space.
This "Big Crunch" effect stems from inefficient particle spreading due to
subdiffusion. We also derive a hierarchy of differential equations for the
moments of the transport process described by the subdiffusive CTRW model. In
the case of an exponential expansion, exact recurrence relations for the
Laplace-transformed moments are obtained. Our results confirm the intuitive
expectation that the medium expansion hinders the mixing of diffusive particles
occupying separate regions. In the case of L\'evy flights, we quantify this
effect by means of the so-called "L\'evy horizon".
