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 1Gyr<~{tau}<~10Gyr 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 1Gyr<~{tau}<~2Gyr. 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 5Gyr<~{tau}<~6Gyr. We propose that USP planets tidally migrated from initial periods in the range 1day<~P<~2days to their observed locations at P<1day over billions of years and trillions of cycles of secular eccentricity excitation and inside-planet damping.