Context. The long-term Global Architecture of Planetary Systems (GAPS) programme has been characterising a sample of young systems with transiting planets via spectroscopic and photometric follow-up observations. One of the main goals of GAPS is measuring planets' dynamical masses and bulk densities to help build a picture of how planets evolve in the early stages of their formation via a comparison between the fundamental physical properties of young and mature exoplanets. Aims. We collected more than 300 high-resolution spectra of the ∼300 Myr old star BD+40 2790 (TOI-2076) over about three years. This star hosts three transiting planets discovered by TESS, with orbital periods of ∼10, 21, and 35 days. From our determined fundamental planetary physical properties, we investigate the temporal evolution of the planetary atmospheres by calculating the expected mass loss rate due to photo-evaporation up to a system age of 5 Gyr. Methods. BD+40 2790 shows an activity-induced scatter larger than 30 m s‑1 in the radial velocities. We employed different methods to measure the stellar radial velocities, along with several models to filter out the dominant stellar activity signal to bring to light the planet-induced signals, which are expected to have semi-amplitudes that are lower by one order of magnitude. We evaluated the mass loss rate of the planetary atmospheres using photo-ionisation hydrodynamic modeling, accounting for the temporal evolution of the stellar high-energy flux through the adoption of different models for X-rays and EUV irradiation. Results. The dynamical analysis confirms that the three sub-Neptune-sized companions (with our radius measurements of Rb = 2.54±0.04, Rc = 3.35±0.05, and Rd = 3.29±0.06 R⊕) have masses that situate them in the planetary regime. We derived 3σ upper limits below or close to the mass of Neptune for all the planets in our sample: 11–12, 12–13.5, and 14–19 M⊕for planets b, c, and d, respectively. In the case of planet d, we found promising clues that the mass could be between ∼7 and 8 M⊕, with a significance level between 2.3–2.5σ (at best). This result must be further investigated using other analysis methods and techniques or using high-precision near-infrared (nIR) spectrographs to collect new radial velocities, which could be less affected by stellar activity. Atmospheric photoevaporation simulations predict that BD+40 2790 b is currently losing its H-He gaseous envelope and that it will be completely lost at an age within 0.5–3 Gyr if its current mass is lower than 12M⊕. Furthermore, BD+40 2790 c could have a lower bulk density than b and might be able to retain its atmosphere up to an age of 5 Gyr. For the outermost object, planet d, we predicted an almost negligible evolution of its mass and radius, induced by photo-evaporation.