The time-variable velocity fields of solar-type stars limit the precision of radial-velocity determinations of their planets' masses, obstructing detection of Earth twins. Since 2015 July, we have been monitoring disc-integrated sunlight in daytime using a purpose-built solar telescope and fibre feed to the HARPS-N stellar radial-velocity spectrometer. We present and analyse the solar radial-velocity measurements and cross-correlation function (CCF) parameters obtained in the first 3 yr of observation, interpreting them in the context of spatially resolved solar observations. We describe a Bayesian mixture- model approach to automated data-quality monitoring. We provide dynamical and daily differential-extinction corrections to place the radial velocities in the heliocentric reference frame, and the CCF shape parameters in the sidereal frame. We achieve a photon-noise-limited radial-velocity precision better than 0.43 m s-1 per 5-min observation. The day-to-day precision is limited by zero-point calibration uncertainty with an RMS scatter of about 0.4 m s-1. We find significant signals from granulation and solar activity. Within a day, granulation noise dominates, with an amplitude of about 0.4 m s-1 and an autocorrelation half-life of 15 min. On longer time-scales, activity dominates. Sunspot groups broaden the CCF as they cross the solar disc. Facular regions temporarily reduce the intrinsic asymmetry of the CCF. The radial-velocity increase that accompanies an active-region passage has a typical amplitude of 5 m s-1 and is correlated with the line asymmetry, but leads it by 3d. Spectral line-shape variability thus shows promise as a proxy for recovering the true radial velocity.