We present the results of a near-infrared (NIR) spectroscopic follow-up survey of 182 M4-L7 low-mass stars and brown dwarfs (BDs) from the BANYAN All-Sky Survey (BASS) for candidate members of nearby, young moving groups (YMGs). We confirm signs of low gravity for 42 new BD discoveries with estimated masses between 8 and 75M_Jup_ and identify previously unrecognized signs of low gravity for 24 known BDs. We refine the fraction of low-gravity dwarfs in the high-probability BASS sample to ~82%. We use this unique sample of 66 young BDs, supplemented with 22 young BDs from the literature, to construct new empirical NIR absolute magnitude and color sequences for low-gravity BDs. We show that low-resolution NIR spectroscopy alone cannot differentiate between the ages of YMGs younger than ~120Myr, and that the BT-Settl atmosphere models do not reproduce well the dust clouds in field or low-gravity L-type dwarfs. We obtain a spectroscopic confirmation of low gravity for 2MASS J14252798-3650229, which is a new ~27M_Jup_, L4 {gamma} bona fide member of AB Doradus. We identify a total of 19 new low-gravity candidate members of YMGs with estimated masses below 13M_Jup_, 7 of which have kinematically estimated distances within 40pc. These objects will be valuable benchmarks for a detailed atmospheric characterization of planetary-mass objects with the next generation of instruments. We find 16 strong candidate members of the Tucana-Horologium association with estimated masses between 12.5 and 14M_Jup_, a regime where our study was particularly sensitive. This would indicate that for this association there is at least one isolated object in this mass range for every 17.5_-5.0_^+6.6^ main-sequence stellar member, a number significantly higher than expected based on standard log-normal initial mass function, however, in the absence of radial velocity and parallax measurements for all of them, it is likely that this over-density is caused by a number of young interlopers from other associations.