We report proper-motion measurements for 427 late-type M, L, and T dwarfs, 332 of which have been measured for the first time. Combining these new proper motions with previously published measurements yields a sample of 841 M7-T8 dwarfs. We combined parallax measurements or calculated spectrophotometric distances, and computed tangential velocities for the entire sample. We find that kinematics for the full and volume-limited 20 pc samples are consistent with those expected for the Galactic thin disk, with no significant differences between late-type M, L, and T dwarfs. Applying an age-velocity relation we conclude that the average kinematic age of the 20 pc sample of ultracool dwarfs is older than recent kinematic estimates and more consistent with age results calculated with population synthesis models. There is a statistically distinct population of high tangential velocity sources (V _tan > 100 km s^–1) whose kinematics suggest an even older population of ultracool dwarfs belonging to either the Galactic thick disk or halo. We isolate subsets of the entire sample, including low surface gravity dwarfs, unusually blue L dwarfs, and photometric outliers in J – K_s color and investigate their kinematics. We find that the spectroscopically distinct class of unusually blue L dwarfs has kinematics clearly consistent with old age, implying that high surface gravity and/or low metallicity may be relevant to their spectral properties. The low surface gravity dwarfs are kinematically younger than the overall population, and the kinematics of the red and blue ultracool dwarfs suggest ages that are younger and older than the full sample, respectively. We also present a reduced proper-motion diagram at 2MASS (Two Micron All Sky Survey) K_s for the entire population and find that a limit of > 18 excludes M dwarfs from the L and T dwarf population regardless of near-infrared color, potentially enabling the identification of the coldest brown dwarfs in the absence of color information.