Aims: We aim to determine the trigonometric parallaxes and proper motions of a sample of ten field L0-L5 dwarfs with spectroscopic evidence for low-gravity atmospheres. The ten sources were located in color-absolute magnitude diagrams and in the Hertzsprung-Russell (HR) diagram for age and mass derivations and were compared with field and star cluster dwarfs of related spectral classification and with state-of-the-art solar-metallicity evolutionary models. Methods: We obtained J and Ks imaging data using 2-4 m class telescopes with a typical cadence of one image per month between 2010 January and 2012 December, in which the data cover a time baseline of nearly three years. We also obtained low resolution optical spectra (R ~ 300, 500-1100 nm) using the 10 m Gran Telescopio de Canarias to assess the presence of lithium absorption in four targets and confirm their young age. The derived parallaxes and proper motions were combined with data from the literature to determine Teff, luminosity, and space velocities. All this information along with the lithium observations was used to assess the ages and masses of the sample. The astrometric curves were also examined for periodic perturbations indicative of unseen companions. Results: Trigonometric parallaxes and proper motions were derived to typical accuracies of a milliarcsecond (mas) and ±10 mas yr-1, respectively. All ten L dwarfs have large motions (μ ≥ 70 mas yr-1) and are located at distances between 9 and 47 pc. They lie above and on the sequence of field dwarfs in the diagrams of absolute J and Ks magnitude versus spectral type and luminosity versus effective temperature, which implies ages similar to or smaller than those typical of the field. In the HR diagram, 2MASS J00332386-1521309 (L4), 2MASS J00452143+1634446 (L2), 2MASS J03552337+1133437 (L5), 2MASS J05012406-0010452 (L4), G 196-3B (L3), 2MASS J17260007+1538190 (L3), and 2MASS J22081363+2921215 (L3) occupy locations that are compatible with the most likely ages in the interval ≈10-500 Myr if they are single objects. All of these dwarfs (except for 2MASS J00332386-1521309) show strong lithium absorption at 670.8 nm, thus confirming the young ages and masses ranging from ≈11 through ≈45 MJup for this subsample. The detection of atomic lithium in the atmosphere of 2MASS J00452143+1634446 (L2) is reported for the first time. The lack of lithium in 2MASS J00332386-1521309 (L4) is not compatible with its position in the HR diagram, suggesting a spectral type earlier than L4. The remaining three dwarfs, 2MASS J02411151-0326587 (L0), 2MASS J10224821+5825453 (L1), and 2MASS J15525906+2948485 (L0) have locations in the HR diagram indicative of older ages and higher masses consistent with the observed lithium depletion previously published. The dynamical studies based on space velocities derived from our parallaxes and proper motions fully support the aforementioned results for 2MASS J00452143+1634446, 2MASS J03552337+1133437, G 196-3B, 2MASS J10224821+5825453, and 2MASS J15525906+2948485. We did not find evidence for the presence of astrometric companions with minimum detectable masses that are typically ≥25 MJup and face-on, circular orbits with periods between 60-90 d and 3 yr around eight targets. Conclusions: The astrometric and spectroscopic data indicate that about 60-70% of the field L-type dwarfs in our sample with evidence for low-gravity atmospheres are indeed young-to-intermediate-age brown dwarfs of the solar neighborhood with expected ages and masses in the intervals ≈10-500 Myr and ≈11-45 MJup. The peaked-shape of the H-band spectra of L dwarfs, a signpost of youth, appears to be present up to ages of 120-500 Myr and intermediate-to-high gravities. Tables 3, 5, Figs. 4, 5, 7, 8, 10, and 11 are available in electronic form at http://www.aanda.org