Radial velocity measurements are presented for 85 late M-and L-type very low mass stars and brown dwarfs obtained with the Magellan Echellette (MagE) spectrograph. Targets primarily have distances within 20 pc of the Sun, with more distant sources selected for their unusual spectral energy distributions. We achieved precisions of 2-3,km s −1 , and combined these with astrometric and spectrophotometric data to calculate U V W velocities. Most are members of the thin disk of the Galaxy, and velocity dispersions indicate a mean age of 5.2±0.2 Gyr for sources within 20 pc. We find signficantly different kinematic ages between late-M dwarfs (4.0±0.2 Gyr) and L dwarfs (6.5±0.4 Gyr) in our sample that are contrary to predictions from prior simulations. This difference appears to be driven by a dispersed population of unusually -2blue L dwarfs which may be more prevalent in our local volume-limited sample than in deeper magnitude-limited surveys. The L dwarfs exhibit an asymmetric U velocity distribution with a net inward flow, similar to gradients recently detected in local stellar samples. Simulations incorporating brown dwarf evolution and Galactic orbital dynamics are unable to reproduce the velocity asymmetry, suggesting non-axisymmetric perturbations or two distinct L dwarf populations. We also find the L dwarfs to have a kinematic age-activity correlation similar to more massive stars. We identify several sources with low surface gravities, and two new substellar candidate members of nearby young moving groups: the astrometric binary DENIS J08230313−4912012AB, a lowprobability member of the β Pictoris Moving Group; and 2MASS J15104786-2818174, a moderate-probability member of the 30-50 Myr Argus Association.All sources were observed with the MagE spectrograph, mounted on the Magellan 6.5m Landon Clay Telescope at Las Campanas Observatory. A complete observing log is given in Table 2. Data were obtained in 15 nights over a 2.5-year period (November 2008 through March 2011) in a variety of seeing and weather conditions. We used the 0. ′′ 7 slit aligned with the parallactic angle, providing 3200-10050Å spectroscopy at an average resolution λ/∆λ ≈ 4100 (∆RV = 73 km s −1 ) and dispersion of ∼0.5Å pixel −1 at 6000Å. Exposure times varied according to source brightness and weather conditions, and ranged from 150-3600 s. Most sources were observed in a single exposure, although a handful were observed in multiple exposures or over multiple nights to improve data quality. In addition to the target, we obtained nightly observations of spectrophotometric standards from Hamuy et al. (1994) for flux calibration. ThAr lamps were observed after each source observation for wavelength calibration, and internal quartz and dome flat field lamps were obtained 2 Source identifications in the text are given in shorthand notation based on the sexigesimal right ascension and declination, Jhhmm±ddmm. Full source names and coordinates are listed in Table 1.