We analyze the parsec-scale jet kinematics from 2007 June to 2018 December of a sample of γ-ray bright blazars monitored roughly monthly with the Very Long Baseline Array (VLBA) at 43 GHz under the VLBA-BU-BLAZAR program. We implement a novel piecewise linear fitting method to derive the kinematics of 521 distinct emission knots from a total of 3705 total intensity images in 22 quasars, 13 BL Lacertae objects, and 3 radio galaxies. Apparent speeds of these components range from 0.01c to 78c, and 18.6% of knots (other than the "core") are quasi-stationary. One-fifth of moving knots exhibit nonballistic motion, with acceleration along the jet within 5 pc of the core (projected) and deceleration farther out. These accelerations occur mainly at locations coincident with quasi-stationary features. We calculate the physical parameters of 273 knots with statistically significant motion, including their Doppler factors, Lorentz factors, and viewing angles. We determine the typical values of these parameters for each jet and the average for each subclass of active galactic nuclei. We investigate the variability of the position angle of each jet over the 10 yr of monitoring. The fluctuations in position of the quasi-stationary components in radio galaxies tend to be parallel to the jet, while no directional preference is seen in the components of quasars and BL Lacertae objects. We find a connection between γ-ray states of blazars and their parsec-scale jet properties, with blazars with brighter 43 GHz cores typically reaching higher γ-ray maxima during flares.