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We investigate the dynamics of optical Airy beams in the one-dimensional fractional Schr\"odinger 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 moir\'e-lattice, hyperbolic-secant, and delta-functional modulation formats. For the HO-moir\'e-lattice potential, it is found that the chirp-free Airy beam experiences multiple spectrum conversions between the Airy and Gaussian patterns in the momentum space, preserving the Airy pattern in the coordinate space. The chirp magnitude of the chirped Airy beam determines whether the spectrum conversion occurs in the momentum space, and the splitting and evolution direction of the beam in the coordinate space. For the HO-hyperbolic-secant potential, the chirp-free Airy beam undergoes spectrum conversion and tunneling, with the positions of the spectrum conversion and tunneling significantly depending on parameters of the hyperbolic-secant potential; however, the spectrum conversion and pattern preservation of the chirped Airy beam occurs only under a certain relation of the chirp and parameters of the potential. In the case of the HO-delta-functional potential, the chirp-free Airy beam experiences abrupt spectrum conversion and a two-step spectrum shift; however, for the chirped Airy beam, the spectrum conversion is affected by the relation between the chirp and height of the potential. Effects of the fractional L\'evy index on the spectrum conversion and pattern preservation of the Airy beams under the action of the three modulation patterns considered here are also explored in detail., Comment: to be published in Chaos, Solitons & Fractals