Context. Gravity modifies the spectral features of young brown dwarfs (BDs). A proper characterization of these objects is crucial for the identification of the least massive and latest-type objects in star-forming regions, and to explain the origin(s) of the peculiar spectrophotometric properties of young directly imaged extrasolar planets and BD companions. Aims: We obtained medium-resolution (R ~ 1500-1700) near-infrared (1.1-2.5 μm) spectra of seven young M9.5-L3 dwarfs classified at optical wavelengths. We aim to empirically confirm the low surface gravity of the objects in the near-infrared. We also test whether self-consistent atmospheric models correctly represent the formation and the settling of dust clouds in the atmosphere of young late-M and L dwarfs. Methods: We used the Infrared Spectrometer And Array Camera (ISAAC) at the Very Large Telescope (VLT) to obtain the spectra of the targets. We compared the spectra to those of mature and young BDs, and to young late-type companions to nearby stars with known ages to identify and study gravity-sensitive features. We computed spectral indices weakly sensitive to the surface gravity to derive near-infrared spectral types. Finally, we found the best fit between each spectrum and synthetic spectra from the BT-Settl 2010 and 2013 atmospheric models. Using the best fit, we derived the atmospheric parameters of the objects and identified which spectral characteristics the models do not reproduce. Results: We confirmed that our objects are young BDs and we found near-infrared spectral types in agreement with the ones determined at optical wavelengths. The spectrum of the L2γ dwarf 2MASSJ232252.99-615127.5 reproduces the spectrum of the planetary mass companion 1RXS J160929.1-210524b well. The BT-Settl models fit the spectra and the 1-5 μm spectral energy distribution of the L0-L3 dwarfs for temperatures between 1600-2000 K. But the models fail to reproduce the shape of the H band and the near-infrared slope of some of our targets. This fact, and the best-fit solutions found with super-solar metallicity, are indicative of a lack of dust, in particular at high altitude, in the cloud models. Conclusions: The modeling of the vertical mixing and of the grain growth will be revised in the next version of the BT-Settl models. These revisions may suppress the remaining non-reproducibilities. Our spectra provide additional templates for the characterization of the numerous young L-type companions that will be detected in the coming years by planet imaging instruments such as VLT/SPHERE, Gemini/GPI, Subaru/SCexAO, and LBTI/LMIRCam. Based on observations made with ESO telescopes at the Paranal Observatory under programs 085.C-0676 and 290.C-5145.Appendices are available in electronic form at http://www.aanda.orgThe spectra (FITS files) are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/564/A55