  We analyze the parsec-scale jet kinematics from 2007 June to 2018 December of
a sample of $\gamma$-ray bright blazars monitored roughly monthly with the Very
Long Baseline Array at 43 GHz under the VLBA-BU-BLAZAR program. We implement a
novel piece-wise 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 non-ballistic 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 ten years 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 $\gamma$-ray states of blazars
and their parsec-scale jet properties, with blazars with brighter 43 GHz cores
typically reaching higher $\gamma$-ray maxima during flares.
