  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-32 b 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 HeI
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.
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$\pm$70)$\times$10$^{11}$ gs$^{-1}$ with a hot (12
400$\pm$2900 K) upper atmosphere; WASP-69 b loses its atmosphere at
(0.9$\pm$0.5)$\times$10$^{11}$ gs$^{-1}$ and 5250$\pm$750 K; and GJ 1214 b,
with a relatively cold outflow of 3750$\pm$750 K, photo-evaporates at
(1.3$\pm$1.1)$\times$10$^{11}$ gs$^{-1}$. 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$\pm$21.5$\times$10$^{11}$
gs$^{-1}$. Our reanalysis of GJ 3470 b yields colder temperatures, 3400$\pm$350
K, but practically the same mass-loss rate as in our previous results. Our
reanalysis of HD 189733 b yields a slightly higher mass-loss rate,
(1.4$\pm$0.5)$\times$10$^{11}$ gs$^{-1}$, and temperature, 12 700$\pm$900 K
compared to previous estimates. Our results support that photo-evaporated
outflows tend to be very light.
