We investigate the dynamics of optical Airy beams in the one-dimensional fractional Schrödinger equation with a harmonic-oscillator (HO) potential subjected to modulation along the propagation distance. Deriving general solutions for propagating beams and particular solutions for Airy waves with/without chirp, we study analytically the spectrum conversion and pattern preservation for the chirp-free and chirped Airy beams in the fractional system including the HO potential with moiré-lattice, hyperbolic-secant, and…