Fast controllable optical passive devices containing intricate couplings of multiple physical fields, for instance, magneto-, electro-, and acousto-optic interactions, are frequently used as critical regulation tools in diverse optical systems. Their unpredictable transient spectral properties under high-speed modulation strongly influence the operational performance of the whole system, but are exhibited very little in conventional spectroscopies due to slow scans with a frame rate of the order of a kilohertz. Here, a dissipative soliton fiber laser assisted dispersive Fourier transform is built to achieve ultrafast spectroscopic characterization. An acoustically induced fiber grating (AIFG), as one typical case of a fast-response device, is studied under driving signals in frequency-switch and frequency-sweep modes. The experimental results reveal the continuous wavelength-sweeping property of the AIFG. Its transmission spectrum for a driving-frequency chirp is destroyed when the sweep time approaches the transit time of the acoustic wave in the grating region. This work opens up an avenue for the measurement of transient physical characteristics of passive devices, such as spectral aberration, transit time, and operational bandwidth.