Various nonequilibrium multi-pulse states can emerge in a mode-locked laser through interactions between the quasi-continuous-wave background (qCWB) and pulses inside the laser cavity. While they have been regarded as unpredictable and hardly controllable due to the noise-like nature of qCWB, we here demonstrate that the qCWB landscape can be manipulated via optoacoustically mediated pulse-to-qCWB interactions, which alters the behaviors of multi-pulse dynamics. In this process, impulsive qCWB modulations are created at well-defined temporal locations, which act as the point emitters and attractive potentials for drifting pulse bunches and soliton rains. Hence, we can transport a single pulse bunch from a certain temporal position to another, and also make soliton rains created and collided exclusively at specific temporal locations. Our study opens up possibilities to control the nonequilibrium multi-pulse phenomena precisely in the time domain, which would offer a practical means of advanced optical information processing.