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We present 1.3- and/or 3-mm continuum images and 3-mm spectral scans,obtained using NOEMA and ALMA, of 21 distant, dusty, star-forming galaxies(DSFGs). Our sample is a subset of the galaxies selected by Ivison et al.(2016) on the basis of their extremely red far-infrared (far-IR) colours andlow {\it Herschel} flux densities; most are thus expected to be unlensed,extraordinarily luminous starbursts at $z \gtrsim 4$, modulo the considerablecross-section to gravitational lensing implied by their redshift. We observed17 of these galaxies with NOEMA and four with ALMA, scanning through the 3-mmatmospheric window. We have obtained secure redshifts for seven galaxies viadetection of multiple CO lines, one of them a lensed system at $z=6.027$ (twoothers are also found to be lensed); a single emission line was detected inanother four galaxies, one of which has been shown elsewhere to lie at$z=4.002$. Where we find no spectroscopic redshifts, the galaxies are generallyless luminous by 0.3-0.4 dex, which goes some way to explaining our failure todetect line emission. We show that this sample contains amongst the mostluminous known star-forming galaxies. Due to their extreme star-formationactivity, these galaxies will consume their molecular gas in $\lesssim 100$Myr, despite their high molecular gas masses, and are therefore plausibleprogenitors of the massive, `red-and-dead' elliptical galaxies at $z \approx3$.

The most distant, luminous, dusty star-forming galaxies: redshifts from NOEMA and ALMA spectral scans