  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.
