  Self-accelerating beams are fascinating solutions of the Schr\"odinger
equation. Thanks to their particular phase engineering, they can accelerate
without the need of external potentials or applied forces. Finite-energy
approximations of these beams have led to many applications, spanning from
particle manipulation to robust in vivo imaging. The most studied and
emblematic beam, the Airy beam, has been recently investigated in the context
of the fractional Schr\"odinger equation. It was notably found that the packet
acceleration would decrease with the reduction of the fractional order. Here, I
study the case of a general nth-order self-accelerating caustic beam in the
fractional Schr\"odinger equation. Using a Madelung decomposition combined with
the wavelet transform, I derive the analytical expression of the beam's
acceleration. I show that the non-accelerating limit is reached for infinite
phase order or when the fractional order is reduced to 1. This work provides a
quantitative description of self-accelerating caustic beams' properties.
