  A fractional diffusion equation with advection term is rigorously derived
from a kinetic transport model with a linear turning operator, featuring a
fat-tailed equilibrium distribution and a small directional bias due to a given
vector field. The analysis is based on bounds derived by relative entropy
inequalities and on two recently developed approaches for the macroscopic
limit: a Fourier-Laplace transform method for spatially homogeneous data and
the so called moment method, based on a modified test function.
