While multiplexed cavities enable rich soliton interactions, achieving deterministic control remains challenging. Here, we report relay-type soliton dynamics in an asynchronous polarization-multiplexed fiber laser, where cavity-induced delayed coupling enforces a sequential energy transfer along the temporal axis and drives solitons into a wave-packet-like intensity envelope. The deviation of the repetition rate from its theoretical value reflects the influence of the dominant path on the overall response of the multiplexed cavity. The resulting wave-packet period provides an experimentally accessible macroscopic timescale to visualize the relay process and reveals an analog translation as a continuous temporal shift of the soliton packet. In the steady regime, delayed coupling and intracavity dissipation, arising from output coupling and fiber losses, establish a dynamic equilibrium that stabilizes the soliton ensemble. Enhanced nonlinearity disrupts this balance and leads to oscillatory behavior. Numerical simulations support the interpretation that delayed coupling acts as an intrinsic optical feedback mechanism and introduces a new degree of freedom for regulating collective soliton dynamics in multiplexed cavities.