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Context. Protostellar jets and outflows are the main outcome of the star formation process, and their analysis can provide us with major clues about the ejection and accretion history of young stellar objects (YSOs). Aims. We aim at deriving the main physical properties of massive jets from near-infrared (NIR) observations, comparing them to those of a large sample of jets from low-mass YSOs, and relating them to the main features of their driving sources. Methods. We present a NIR imaging (H2 and Ks) and low-resolution spectroscopic (0.95-2.50 µm) survey of 18 massive jets towards GLIMPSE extended green objects (EGOs), driven by intermediate- and high-mass YSOs, which have bolometric luminosities (Lbol) between 4 × 10² and 1.3 × 105 L⊙. Results. As in low-mass jets, H2 is the primary NIR coolant, detected in all the analysed flows, whereas the most important ionic tracer is [Fe?ii], detected in half of the sampled jets. Our analysis indicates that the emission lines originate from shocks at high temperatures and densities. No fluorescent emission is detected along the flows, regardless of the source bolometric luminosity. On average, the physical parameters of these massive jets (i.e. visual extinction, temperature, column density, mass, and luminosity) have higher values than those measured in their low-mass counterparts. The morphology of the H2 flows is varied, mostly depending on the complex, dynamic, and inhomogeneous environment in which these massive jets form and propagate. All flows and jets in our sample are collimated, showing large precession angles. Additionally, the presence of both knots and jets suggests that the ejection process is continuous with burst episodes, as in low-mass YSOs. We compare the flow H2 luminosity with the source bolometric luminosity confirming the tight correlation between these two quantities. Five sources, however, display a lower LH2/Lbol efficiency, which might be related to YSO evolution. Most important, the inferred LH2 vs. Lbol relationship agrees well with the correlation between the momentum flux of the CO outflows and the bolometric luminosities of high-mass YSOs indicating that outflows from high-mass YSOs are momentum driven, as are their low-mass counterparts. We also derive a less stringent correlation between the inferred mass of the H2 flows and Lbol of the YSOs, indicating that the mass of the flow depends on the driving source mass. Conclusions. By comparing the physical properties of jets in the NIR, a continuity from low- to high-mass jets is identified. Massive jets appear as a scaled-up version of their low-mass counterparts in terms of their physical parameters and origin. Nevertheless, there are consistent differences such as a more variegated morphology and, on average, stronger shock conditions, which are likely due to the different environment in which high-mass stars form. [ABSTRACT FROM AUTHOR]