Neutral hydrogen (HI) velocity dispersions are believed to be set by turbulence in the interstellar medium (ISM). Although turbulence is widely believed to be driven by star formation (SF), recent studies have shown that this driving mechanism may not be dominant in regions of low SF rate surface density (SFRSD), such as found in dwarf galaxies or the outer regions of spirals. We have generated average HI line profiles in a number of nearby dwarfs and low-mass spirals by co-adding HI spectra in regions with either a common radius or SFRSD. We find that the spatially-resolved superprofiles are composed of a central narrow peak (5-15 km/s) with higher velocity wings to either side. With the assumption that the central peak reflects the turbulent velocity dispersion, we compare HI kinematics to local ISM properties, including surface mass densities and measures of SF. The HI velocity dispersion is correlated most strongly with surface mass density, which points at a gravitational origin for turbulence, but it is unclear which instabilities can operate efficiently in these systems. SF energy is produced at a level sufficient to drive HI turbulent motions where SFRSD > 10^-4 Msun yr^-1 kpc^-2. At low SF intensities, SF does not supply enough energy for turbulence, nor does it uniquely determine the velocity dispersion. Nevertheless, SF appears to provide a lower threshold for HI velocity dispersions. We find that coupling efficiency decreases with increasing SFRSD, consistent with a picture where SF couples to the ISM with constant efficiency, but that less of that energy is found in HI at higher SFRSD. We examine a number of potential drivers of HI turbulence, including SF, gravitational instabilities, the magnetorotational instability, and accretion, and find that no single mechanism can drive the observed levels of turbulence at low SFRSD. We discuss possible solutions to this conundrum.
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