  Exoplanet systems are thought to evolve on secular timescales over billions
of years. This evolution is impossible to directly observe on human timescales
in most individual systems. While the availability of accurate and precise age
inferences for individual exoplanet host stars with ages $\tau$ in the interval
$1~\text{Gyr}\lesssim~\tau~\lesssim10~\text{Gyr}$ would constrain this
evolution, accurate and precise age inferences are difficult to obtain for
isolated field dwarfs like the host stars of most exoplanets. The Galactic
velocity dispersion of a thin disk stellar population monotonically grows with
time, and the relationship between age and velocity dispersion in a given
Galactic location can be calibrated by a stellar population for which accurate
and precise age inferences are possible. Using a sample of subgiants with
precise age inferences, we calibrate the age--velocity dispersion relation in
the Kepler field. Applying this relation to the Kepler field's planet
populations, we find that Kepler-discovered systems plausibly in second-order
mean-motion resonances have $1~\text{Gyr}\lesssim~\tau~\lesssim2~\text{Gyr}$.
The same is true for systems plausibly in first-order mean-motion resonances,
but only for systems likely affected by tidal dissipation inside their
innermost planets. These observations suggest that many planetary systems
diffuse away from initially resonant configurations on secular timescales. Our
calibrated relation also indicates that ultra-short-period (USP) planet systems
have typical ages in the interval
$5~\text{Gyr}\lesssim~\tau~\lesssim6~\text{Gyr}$. We propose that USP planets
tidally migrated from initial periods in the range
$1~\text{d}\lesssim~P~\lesssim2~\text{d}$ to their observed locations at
$P<1~\text{d}$ over billions of years and trillions of cycles of secular
eccentricity excitation and inside-planet damping.
