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We report the results of the photodynamical analyses of four compact, tighttriple stellar systems, KICs 6964043, 5653126, 5731312, 8023317, based largelyon $Kepler$ and $TESS$ data. All systems display remarkable eclipse timing andeclipse depth variations, the latter implying a non-aligned outer orbit.Moreover, KIC 6964043 is also a triply eclipsing system. We combinedphotometry, ETV curves, and archival spectral energy distribution data toobtain the astrophysical parameters of the constituent stars and the orbitalelements with substantial precision. KICs 6964043 and 5653126 were found to benearly flat with mutual inclinations $i_{mut}=4.1$ deg and $12.3$ deg,respectively, while KICs 5731312, 8023317 ($i_{mut}=39.4$ deg and $55.7$ deg,respectively) are found to lie in the high $i_{mut}$ regime of the vonZeipel-Kozai-Lidov (ZKL) theorem. We show that currently both high inclinationtriples exhibit observable unusual retrograde apsidal motion. Moreover, theeclipses will disappear in all but one of the four systems within a fewdecades. Short-term numerical integrations of the dynamical evolution revealthat both high inclination triples are currently subject to ongoing, largeamplitude ($\Delta e\sim0.3$) inner eccentricity variations on centuries-longtimescales, in accord with the ZKL theorem. Longer-term integrations predictthat two of the four systems may become dynamically unstable on $\sim$ Gyrtimescales, while in the other two triples common envelope phases and stellarmergers may occur. Finally we investigate the dynamical properties of a sampleof 71 KIC/TIC triples statistically, and find that the mutual inclinations andouter mass ratios are anti-correlated at the 4$\sigma$ level. We discuss theimplications for the formation mechanisms of compact triples.

Von Zeipel - Lidov - Kozai cycles in action: $Kepler$ triples with eclipse depth variations: KICs 6964043, 5653126, 5731312 and 8023317