We present a class of diffraction-free partially coherent beams, each member of which comprises a finite-power, non-accelerating Airy bump residing on a statistically homogeneous, Gaussian-correlated background. We examine free-space propagation of soft apertured realizations of the proposed beams and show that their evolution is governed by two spatial scales: the coherence width of the background and the aperture size. A relative magnitude of these factors determines the practical range of propagation distances over which the novel beams can withstand diffraction. The proposed beams can find applications to imaging and optical communications through random media.