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Forming high-mass stars have a significant effect on their natal environment. Their feedback pathways, including winds, outflows, and ionising radiation, shape the evolution of their surroundings which impacts the formation of the next generation of stars. They create or reveal dense pillars of gas and dust towards the edges of the cavities they clear. They are modelled in feedback simulations, and the sizes and shapes of the pillars produced are consistent with those observed. However, these models predict measurably different kinematics which provides testable discriminants. Here we present the first ALMA Compact Array (ACA) survey of 13 pillars in Carina, observed in $^{12}$CO, $^{13}$CO and C$^{18}$O J=2-1, and the 230 GHz continuum. The pillars in this survey were chosen to cover a wide range in properties relating to the amount and direction of incident radiation, proximity to nearby irradiating clusters and cloud rims, and whether they are detached from the cloud. With these data, we are able to discriminate between models. We generally find pillar velocity dispersions of $<$ 1 km s$^{-1}$ and that the outer few layers of molecular emission in these pillars show no significant offsets from each other, suggesting little bulk internal motions within the pillars. There are instances where the pillars are offset in velocity from their parental cloud rim, and some with no offset, hinting at a stochastic development of these motions., Comment: 24 Pages, 19 figures. Accepted to MNRAS