We report the 3D kinematics of 27 Mira-like stars in the northern, eastern, and southern periphery of the Large Magellanic Cloud (LMC), based on Gaia proper motions and a dedicated spectroscopic follow-up. Low-resolution spectra were obtained for more than 40 Mira-like candidates, selected to trace known substructures in the LMC periphery. Radial velocities and stellar parameters were derived for all stars. Gaia data release 3 astrometry and photometry were used to discard outliers, derive periods for those stars with available light curves, and determine their photometric chemical types. The 3D motion of the stars in the reference frame of the LMC revealed that most of the stars, in all directions, have velocities consistent with being part of the LMC disc population, out of equilibrium in the radial and vertical directions. A suite of numerical simulations was used to constrain the most likely past interaction history between the Clouds given the phase-space distribution of our targets. Model realizations in which the Small Magellanic Cloud (SMC) had three pericentric passages around the LMC best resemble the observations. The interaction history of those model realizations has a recent SMC pericentric passage (~320 Myr ago), preceded by an SMC crossing of the LMC disc at ~0.97 Gyr ago, having a radial crossing distance of only ~4.5 kpc. The previous disc crossing of the SMC was found to occur at ~1.78 Gyr ago, with a similar radial crossing distance of ~5.6 kpc.