  We present a survey of the molecular gas in 61 submillimetre galaxies (SMGs)
selected from 870$\mu$m continuum surveys of the COSMOS, UDS and ECDFS fields,
using the Atacama Large Millimeter Array (ALMA) and the Northern Extended
Millimeter Array (NOEMA). 46 $^{12}$CO ($J=$2-5) emission lines are detected in
45 of the targets at $z=$1.2-4.8, with redshifts indicating that those which
are submillimetre bright and undetected/faint in the optical/near-infrared
typically lie at higher redshifts, with a gradient of $\Delta z/\Delta
S_{870}=$0.11$\pm$0.04mJy$^{-1}$. We also supplement our data with literature
sources to construct a statistical CO spectral line energy distribution and
find the $^{12}$CO line luminosities in SMGs peak at $J_{\rm up}\sim$6,
consistent with the Cosmic Eyelash, among similar studies. Our SMGs lie mostly
on or just above the main sequence, displaying a decrease in their gas
depletion timescales $t_{\rm dep} = M_{\rm gas}/{\rm SFR}$ with redshift in the
range $z\sim$1-5 and a median of 200$\pm$50Myr at $z\sim$2.8. This coincides
with an increase in molecular gas fraction $\mu_{\rm gas} = M_{\rm gas}/M_\ast$
across the same redshift range. Finally we demonstrate that the $M_{\rm
baryon}$-$\sigma$ distribution of our SMGs is consistent with that followed by
early-type galaxies in the Coma cluster, providing strong support to the
suggestion that SMGs are progenitors of massive local spheroidal galaxies. On
the basis of this we suggest that the SMG populations above and below an
870-$\mu$m flux limit of $S_{870}\sim$5mJy may correspond to the division
between slow- and fast-rotators seen in local early-type galaxies.
