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Recently, large samples of visually classified early-type galaxies (ETGs) containing dust have been identified using space-based infrared observations with the Herschel Space Telescope. The presence of large quantities of dust in massive ETGs is peculiar as X-ray haloes of these galaxies are expected to destroy dust in ~107 yr (or less). This has sparked a debate regarding the origin of the dust: Is it internally produced by asymptotic giant branch stars, or is it accreted externally through mergers? We examine the 2D stellar and ionized gas kinematics of dusty ETGs using integral field spectroscopy observations from the SAMI Galaxy Survey, and integrated star formation rates, stellar masses and dust masses from the GAMA survey. Only 8 per cent (4/49) of visually classified ETGs are kinematically consistent with being dispersion-supported systems. These 'dispersion-dominated galaxies' exhibit discrepancies between stellar and ionized gas kinematics, either offsets in the kinematic position angle or large differences in the rotational velocity, and are outliers in star formation rate at a fixed dust mass compared to normal star-forming galaxies. These properties are suggestive of recent merger activity. The remaining ~90 per cent of dusty ETGs have low velocity dispersions and/or large circular velocities, typical of 'rotation-dominated galaxies'. These results, along with the general evidence of published works on X-ray emission in ETGs, suggest that they are unlikely to host hot, X-ray gas consistent with their low M* when compared to dispersiondominated galaxies. This means that dust will be long-lived and thus these galaxies do not require external scenarios for the origin of their dust content.