A crucial property of the substorm growth phase is the current sheet thinning, which is often attributed to adiabatic convection. Injecting low-entropy bubbles reduce pressure-balance inconsistencies and restore current sheet thickness to its initial value. Recent observations from Cluster and MMS showed additional thinning of the current sheet in the wake of a transitory bubble injection compared with the configuration before the injection. We employ the Rice Convection Model-MHD coupled code to investigate how the transport of bubbles causes the fast thinning. The simulation results reconstructed the observations' most prominent characteristics. We find more earthward transport of the magnetic flux in the bubble than its wake, therefore depleting magnetic flux and stretching the magnetic field lines there. Interestingly, additional R0-sense field-aligned currents close the enhanced dawn-to-dusk current behind the bubble.