Dissipative solitons (DSs) provide a prototypical platform for studying nonequilibrium dynamics governed by the interplay between nonlinearity, dissipation, and external driving. In particular, subjected to external potentials, DSs exhibit a rich variety of nonlinear dynamical behaviors. This work investigates DS dynamics in a moving intracavity potential generated by phase modulation whose frequency is detuned from an integer multiple of the cavity repetition rate. Modulation detuning introduces a timescale mismatch between the DS and the moving potential, leading to two distinct dynamical regimes. At large detuning, the DS cannot continuously track the moving potential and exhibits persistent spectral oscillations and temporal meandering, whereas at sufficiently small detuning it enters a synchronized locking state. Furthermore, both the oscillation period and amplitude can be tuned via the modulation detuning and modulation depth. These results reveal how asynchronous driving reshapes DS dynamics and establish a route for controlling their dynamical response in nonlinear optical cavities.