  The propagation of Airy beams in free space is characterized by being non
dispersive, which warrants the shape invariance of their intensity
distribution, and self-accelerating along the transverse direction. These
distinctive traits are still present in partially coherent Airy beams as long
as the reach of their back tail (and hence their energy content) is not
importantly reduced. To investigate the effects associated with the decrease of
the beam coherence and its power content (by smoothly reducing the reach of
their back tails), here we introduce a novel and insightful methodology based
on a generalization of the concept of flux trajectory for paraxial partially
coherent beams. This methodologies emphasizes the role of phase relations, thus
helping to clarify why and how the beam smears out spatially along its
propagation. This formalism, though, is general enough to tackle other types of
structured light beams with whatever degree of partial coherence, from full
coherence to total incoherence.
