Characterisation of atmospheres undergoing photo-evaporation is key to understanding the formation, evolution, and diversity of planets. However, only a few upper atmospheres that experience this kind of hydrodynamic escape have been characterised. Our aim is to characterise the upper atmospheres of the hot Jupiters HAT-P-32b and WASP-69 b, the warm sub-Neptune GJ 1214 b, and the ultra-hot Jupiter WASP-76 b through high-resolution observations of their He I triplet absorption. In addition, we also reanalyse the warm Neptune GJ 3470 b and the hot Jupiter HD 189733 b. We used a spherically symmetric 1D hydrodynamic model coupled with a non-local thermodynamic equilibrium model for calculating the He I triplet distribution along the escaping outflow. Comparing synthetic absorption spectra with observations, we constrained the main parameters of the upper atmosphere of these planets and classify them according to their hydrodynamic regime. Our results show that HAT-P-32 b photo-evaporates at (130 ± 70) ×1011 g s−1 with a hot (12 400 ± 2900 K) upper atmosphere; WASP-69 b loses its atmosphere at (0.9 ± 0.5) ×1011 g s−1 and 5250 ± 750 K; and GJ 1214b, with a relatively cold outflow of 3750 ± 750 K, photo-evaporates at (1.3 ± 1.1) ×1011 g s−1. For WASP-76 b, its weak absorption prevents us from constraining its temperature and mass-loss rate significantly; we obtained ranges of 6000-17 000 K and 23.5 ± 21.5 ×1011 g s−1. Our reanalysis of GJ 3470 b yields colder temperatures, 3400 ± 350 K, but practically the same mass-loss rate as in our previous results. Our reanalysis of HD 189733 b yields a slightly higher mass-loss rate, (1.4 ± 0.5) × 1011 g s−1, and temperature, 12 700 ± 900 K compared to previous estimates. We also found that HAT-P-32 b, WASP-69 b, and WASP-76 b undergo hydrodynamic escape in the recombination-limited regime, and that GJ 1214 b is in the photon-limited regime. Our results support that photo-evaporated outflows tend to be very light, H/He ≳ 98/2. The dependences of the mass-loss rates and temperatures of the studied planets on the respective system parameters (X-ray and ultraviolet stellar flux, gravitational potential) are well explained by the current hydrodynamic escape models.