Context. Jets from young stars can be highly asymmetric and have multiple velocity components. Aims. To clarify the origin of jet asymmetries and constrain the launch mechanism, we study as a test case the physical and kinematical structure of the prototypical asymmetric flow emitted by DG Tau B. Methods. The analysis of deep, high spectral resolution observations taken with the KECK telescope allows us to infer the properties and the spatial distribution of the velocity components in the two jet lobes. From selected line ratios we derive the gas physical conditions (the electron and total density, n e and n H , the ionisation fraction, x e , and the temperature, T e ), as a function of the distance from the source and the gas velocity. The presence of dust grains in the jet is investigated by estimating the gas-phase abundance of calcium with respect to its solar value. Results. The detected lines show broad velocity profiles at the base of the jet (up to ∼100 km s −1 ), where up to three velocity components are detected. At 5 from the source, however, only the denser and more excited high-velocity components survive and the lines are narrower (∼10-30 km s −1 ). The jet is strongly asymmetric in the velocity and in its physical structure. The red lobe, which is slower (∼140 km s −1 ) and more collimated (opening angle: α ∼ 3-4 • ), presents low ionisation fractions (x e ∼ 0.1-0.4) and temperatures (T e < 5 × 10 3 K), while the total density is up to ∼2.5 × 10 4 cm −3 . The blue lobe, faster (∼-320 km s −1 ) and less collimated (α ∼ 14 • ), is also less dense (n H < 10 4 cm −3 ), but highly excited (T e up to ∼5 × 10 4 K and x e up to 0.9). The estimated mass-loss rate turns out to be similar in the two lobes (∼6-8 × 10 −9 M yr −1 ), while the flux of the linear momentum is three times higher in the blue one (∼2.5 × 10 −7 M yr −1 km s −1 ). Calcium is strongly depleted with respect to its solar abundance, indicating that the jet contains dust grains. The depletion is lower for higher velocities, which is consistent with dust destruction by shocks. Conclusions. The similar mass-loss rate in the two lobes suggests that the ejection power is comparable on the two sides of the system, as expected from a magneto-centrifugal ejection mechanism, and that the observed asymmetries are caused by a different mass load and propagation properties in an inhomogeneous environment. The presence of dust grains implies that the jet is generated from a region of the disc extending beyond the dust sublimation radius.