We analyze the parsec-scale jet kinematics of a sample of gamma-ray bright blazars monitored roughly monthly with the Very Long Baseline Array at 43 GHz from 2007 to 2018. We implement a novel piece-wise linear fit to the trajectories of over 500 distinct emission features (knots) in the jets of 23 quasars, 12 BL Lacertae objects, and 3 radio galaxies. The apparent speeds range from < 1c to ~50 c; ~25% of the knots are quasi-stationary. We assess the stability of the jet direction of each source; 9 sources show a change in the jet position angle over time. Approximately 20% of moving knots exhibit non-ballistic motion, defined as deviations from a pure linear trajectory using a chi-square test. These deviations take the form of discrete regions of acceleration, with a net positive acceleration along the jet direction within 3 pc (projected) of the mm-wave "core" and deceleration farther down the jet. Acceleration perpendicular to the average jet direction only occurs within the first 2 pc of the mm-wave core. We also show that the regions of acceleration are associated with the locations of the quasi-stationary features. We derive the physical parameters of over 450 superluminal motion segments using the apparent speeds and timescales of flux variability. These parameters include variability Doppler factors, Lorentz factors, and viewing angles - crucial for modeling the gamma-ray emission in blazars. This research is supported by NASA through the Fermi Guest Investigator Program grants 80NSSC17K0649 and 80NSSC20K1567, and the NRAO Student Observing Support Program.