In the centres of the Milky Way and M83, the global environmental properties thought to control star formation are indistinguishable within observational uncertainties. Despite this, present-day star formation rates in the centres of each galaxy differ by an order of magnitude. In this thesis, I explore the gas kinematics of both regions to understand the origin of this difference. In Chapter 2, I present an overview and data release of the spectral line component of the SMA Large Program, CMZoom. The CMZoom survey targeted 10 dense gas and shock tracers in all gas within the Central Molecular Zone (CMZ; inner few hundred pc of the Galaxy) above a column density of N(H2) >= 10^23 cm^-2, between an observing frequency of 216 -232 GHz, including three CO isotopologues, multiple transitions of H2CO, SiO, OCS and SO. I extracted spectra from all compact 230 GHz continuum sources (cores) in the previously published CMZoom catalogue and fit line profiles to the spectra. After detailed quality controls, I used the fit results from the H2CO 3(0,3)-2(0,2) transition -- which represents 88.8% of the total mass of CMZoom cores -- to determine the core kinematic properties. I find that only four cores are self-gravitating, but that the remainder are consistent with being in hydrostatic equilibrium and confined by the high external pressure in the Galactic Centre. Using star formation tracer associations from an forthcoming publication, I estimate a present-day star formation rate of 0.009 M yr^-1 for all cores in the Galactic Centre with an H2CO detection. I find that the line ratios of CMZoom cores in the Galactic Centre are indistinguishable from CMZoom cores which lie outside the Galactic Centre. This suggests that the chemical differences observed between the CMZ and disk clouds does not propagate down to core scales. I find only two convincing proto-stellar outflows throughout the survey, ruling out the possibility of a previously undetected population of forming high-mass stars. Despite having sufficient sensitivity and resolution, in the large CMZoom survey area I find no high-velocity compact clouds (HVCCs) which have been claimed as evidence for intermediate mass black holes interacting with molecular gas clouds. In Chapter 3, I use ALMA observations of HCN (1-0) and HCO+ (1-0) to trace the dense gas at the size scale of individual molecular clouds (0.54"', 12pc) in the inner ~ 500 pc of M83, and compare this to gas clouds at similar resolution and galactocentric radius in the Milky Way. I find that both the overall gas distribution and the properties of individual clouds are very similar in the two galaxies, and that a common mechanism may be responsible for instigating star formation in both circumnuclear rings. Given the remarkable similarity in gas properties, the most likely explanation for the order of magnitude difference in SFR is time variability, with the Central Molecular Zone (CMZ) currently being at a more quiescent phase of its star formation cycle. I show M83's SFR must have been an order of magnitude higher 5-7 Myr ago. M83's `starburst' phase was highly localised, both spatially and temporally, greatly increasing the feedback efficiency and ability to drive galactic-scale outflows. This highly dynamic nature of star formation and feedback cycles in galaxy centres means (i) modeling and interpreting observations must avoid averaging over large spatial areas or timescales, and (ii) understanding the multi-scale processes controlling these cycles requires comparing snapshots of a statistical sample of galaxies in different evolutionary stages