  We use ALMA observations of four sub-millimetre galaxies (SMGs) at $z\sim2-3$
to investigate the spatially resolved properties of the inter-stellar medium
(ISM) at scales of 1--5 kpc (0.1--0.6$''$). The velocity fields of our sources,
traced by the $^{12}$CO($J$=3-2) emission, are consistent with disk rotation to
first order, implying average dynamical masses of
$\sim$3$\times10^{11}$M$_{\odot}$ within two half-light radii. Through a
Bayesian approach we investigate the uncertainties inherent to dynamically
constraining total gas masses. We explore the covariance between the stellar
mass-to-light ratio and CO-to-H$_{2}$ conversion factor, $\alpha_{\rm CO}$,
finding values of $\alpha_{\rm CO}=1.1^{+0.8}_{-0.7}$ for dark matter fractions
of 15 \%. We show that the resolved spatial distribution of the gas and dust
continuum can be uncorrelated to the stellar emission, challenging energy
balance assumptions in global SED fitting. Through a stacking analysis of the
resolved radial profiles of the CO(3-2), stellar and dust continuum emission in
SMG samples, we find that the cool molecular gas emission in these sources
(radii $\sim$5--14 kpc) is clearly more extended than the rest-frame $\sim$250
$\mu$m dust continuum by a factor $>2$. We propose that assuming a constant
dust-to-gas ratio, this apparent difference in sizes can be explained by
temperature and optical-depth gradients alone. Our results suggest that caution
must be exercised when extrapolating morphological properties of dust continuum
observations to conclusions about the molecular gas phase of the ISM.
