We report on the growth by solid-source MBE of random-alloy AlxIn1-xAsySb1-y layers lattice-matched to (001)-GaSb substrates, with x(Al) is an element of [0.25; 0.75]. The samples quality and morphology were characterized by X-ray diffraction, Nomarski microscopy and atomic force microscopy. Layers grown at 400 degrees C demonstrated smooth surfaces and no sign of phase decomposition. Samples with x(Al) <= 0.60 demonstrated photoluminescence (PL) at 300 K whereas samples with higher Al content only demonstrated PL at low temperature. Samples grown at 430 degrees C, in contrast, exhibited PL at low temperature only, whatever their composition. Inferred bandgap energies corroborate the estimation of a non-null quaternary bowing parameter made by Donati, Kaspi and Malloy in Journal of Applied Physics 94 (2003) 5814. Upon annealing, the PL peak energies increased, getting even closer to the theoretical values. These results are in agreement with recently published results on digital AlInAsSb alloys. Our work, which reports the first evidence for PL emission from random-alloy AlInAsSb layers lattice-matched to GaSb, opens the way to their use in optoelectronic devices. (C) 2017 Elsevier B.V. All rights reserved.