  The rotational evolution of cool dwarfs is poorly constrained after around
1-2 Gyr due to a lack of precise ages and rotation periods for old
main-sequence stars. In this work we use velocity dispersion as an age proxy to
reveal the temperature-dependent rotational evolution of low-mass Kepler
dwarfs, and demonstrate that kinematic ages could be a useful tool for
calibrating gyrochronology in the future. We find that a linear gyrochronology
model, calibrated to fit the period-Teff relationship of the Praesepe cluster,
does not apply to stars older than around 1 Gyr. Although late-K dwarfs spin
more slowly than early-K dwarfs when they are young, at old ages we find that
late-K dwarfs rotate at the same rate or faster than early-K dwarfs of the same
age. This result agrees qualitatively with semi-empirical models that vary the
rate of surface-to-core angular momentum transport as a function of time and
mass. It also aligns with recent observations of stars in the NGC 6811 cluster,
which indicate that the surface rotation rates of K dwarfs go through an epoch
of inhibited evolution. We find that the oldest Kepler stars with measured
rotation periods are late-K and early-M dwarfs, indicating that these stars
maintain spotted surfaces and stay magnetically active longer than more massive
stars. Finally, based on their kinematics, we confirm that many rapidly
rotating GKM dwarfs are likely to be synchronized binaries.
