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Runaway OB stars are ejected from their parent clusters via two mechanisms, both involving multiple stars: the dynamical ejection scenario (DES) and the binary supernova scenario (BSS). We constrain the relative contributions from these two ejection mechanisms in the Small Magellanic Cloud (SMC) using data for 304 field OB stars from the spatially complete, Runaways and Isolated O-Type Star Spectroscopic Survey of the SMC (RIOTS4). We obtain stellar masses and projected rotational velocities $v_r\sin i $ for the sample using RIOTS4 spectra, and use transverse velocities $v_{\rm loc}$ from $\it{Gaia}$ DR2 proper motions. Kinematic analyses of the masses, $v_r\sin i $, non-compact binaries, high-mass X-ray binaries, and Oe/Be stars largely support predictions for the statistical properties of the DES and BSS populations. We find that dynamical ejections dominate over supernova ejections by a factor of $\sim 2-3$ in the SMC, and our results suggest a high frequency of DES runaways and binary ejections. Objects seen as BSS runaways also include two-step ejections of binaries that are reaccelerated by SN kicks. We find that two-step runaways likely dominate the BSS runaway population. Our results further imply that any contribution from $\it{in-situ}$ field OB star formation is small. Finally, our data strongly support the post-mass-transfer model for the origin of classical Oe/Be stars, providing a simple explanation for the bimodality in the $v_r\sin i $ distribution and high, near-critical, Oe/Be rotation velocities. The close correspondence of Oe/Be stars with BSS predictions implies that the emission-line disks are long-lived.