Based on multi-satellite and ground observations, we investigated an influence of long-lasting and gradual enhancements of magnetic flux transport rate on the magnetotail fast flow duration. On March 10th, 2009, THEMIS-B, which was located in the central plasma sheet of middle distant magnetotail (XGSM ∼−25.8 RE), observed the fast flows with the velocity exceeding 300 km/s, lasting over three hours for intense southward Interplanetary Magnetic Field (IMF) period. During long-lasting fast flows, AL index variations were very extensive and their recovery was much slow. Pi 2 waves were observed at the ground observatories around the THEMIS's footpoints and at low-/mid-latitudes. The aspect for these ALAL variations suggests Steady Magnetospheric Convection (SMC), but clear substorm signatures were also observed. Further magnetic dipolarization was detected by THEMIS-A at XGSM ∼−8.2 RE and its nearby THEMIS-E. Only THEMIS-A observed associated energetic electron flux enhancements. Therefore, the fast flows occurred during substorm with energetic particle injections at “imitative” SMC, which would be driven by prolonged intense southward IMF. The cumulative transport rates of magnetic and Poynting fluxes consecutively and gradually enhanced. On the other hand, THEMIS-C detected much shorter fast flows with the duration of 37 minutes at XGSM ∼−18.1 RE and weak/gradual substorm-associated dipolarization. However, the cumulative magnetic flux transport rate enhanced only during the fast flow interval and was saturated after the fast flows. From different magnetic transport rate profiles at THEMIS-B and THEMIS-C, the realms of dipolar-configured field lines expanded to near THEMIS-C's position responsible for long-lasting fast flow-associated consecutive and gradual magnetic flux pileup. Because resultant “high-speed flow braking” region was retreated into a few RE tailward direction, long-lasting fast flows were almost stemmed. These results suggest that the cumulative magnetic flux transport rate is one of the important factors to determine “fate” (duration) of the magnetotail fast flows.