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We present a catalog of ~100,000 periodic variable stars in TESS FFI dataamong members of widely distributed moving groups identified with Gaia in theprevious papers in the series. By combining the periods from our catalogattributable to rotation with previously derived rotation periods for benchmarkopen clusters, we develop an empirical gyrochronology relation of angularmomentum evolution that is valid for stars with ages 10-1000 Myr. Excludingstars rotating faster than 2 days, which we find are predominantly binaries, weachieve a typical age precision of ~0.2-0.3 dex and improving at older ages.Importantly, these empirical relations apply to not only FGK-type stars butalso M-type stars, due to the angular momentum distribution being muchsmoother, simpler, continuous and monotonic as compared to the rotation perioddistribution. As a result, we are also able to begin tracing in fine detail thenature of angular momentum loss in low-mass stars as functions of mass and age.We characterize the stellar variability amplitudes of the cool stars asfunctions of mass and age, which may correlate with the starspot coveringfractions. We also identify pulsating variables among the hotter stars in thecatalog, including $\delta$ Scuti, $\gamma$ Dor and SPB-type variables. Thesedata represent an important step forward in being able to estimate precise agesof FGK- and M-type stars in the field, starting as early as thepre-main-sequence phase of evolution.

Untangling the Galaxy. IV. Empirical Constraints on Angular Momentum Evolution and Gyrochronology for Young Stars in the Field