We present VLT/X-shooter observations of a sample of 36 accreting low-mass stellar and substellar objects (YSOs) in the Lupus star-forming region, spanning a range in mass from ~0.03 to ~1.2 M⊙, but mostly with 0.1 M⊙<M⋆< 0.5 M⊙. Our aim is twofold: firstly, to analyse the relationship between excess-continuum and line emission accretion diagnostics, and, secondly, to investigate the accretion properties in terms of the physical properties of the central object. The accretion luminosity (Lacc), and in turn the accretion rate (Ṁacc), was derived by modelling the excess emission from the UV to the near-infrared as the continuum emission of a slab of hydrogen. We computed the flux and luminosity (Lline) of many emission lines of H  , He  , and Ca   ii, observed simultaneously in the range from ~330 nm to 2500 nm. The luminosity of all the lines is well correlated with Lacc. We provide empirical relationships between Lacc and the luminosity of 39 emission lines, which have a lower dispersion than relationships previously reported in the literature. Our measurements extend the Paβ and Brγ relationships to Lacc values about two orders of magnitude lower than those reported in previous studies. We confirm that different methodologies of measuring Lacc and Ṁacc yield significantly different results: Hα line profile modelling may underestimate Ṁacc by 0.6 to 0.8 dex with respect to Ṁacc derived from continuum-excess measures. These differences may explain the probably spurious bi-modal relationships between Ṁacc and other YSOs properties reported in the literature. We derived Ṁacc in the range 2 × 10-12–4 × 10-8 M⊙ yr-1 and conclude that Ṁacc ∝ M⋆1.8(±0.2), with a dispersion lower by a factor of about 2 than in previous studies. A number of properties indicate that the physical conditions of the accreting gas are similar over more than 5 orders of magnitude in Ṁacc, confirming previous suggestions that the geometry of the accretion flow controls the rate at which the disc material accretes onto the central star.