In order to understand which process (e.g. galactic winds, cold accretion) is responsible for the cool (T ∼ 104 K) halo gas around galaxies, we embarked on a programme to study the star formation properties of galaxies selected by their Mg ii absorption signature in quasar spectra. Specifically, we searched for the Hα line emission from galaxies near very strong z ≃ 2 Mg ii absorbers (with rest-frame equivalent width Å) because these could be the signposts of outflows or inflows. Surprisingly, we detect Hα from only four hosts out of 20 sightlines (and two out of the 19 H i-selected sightlines), despite reaching a star formation rate (SFR) sensitivity limit of 2.9 M⊙ yr−1 (5σ) for a Chabrier initial mass function. This low success rate (4/20) is in contrast with our z ≃ 1 survey where we detected 66 per cent (14/21) of the Mg ii hosts (down to 0.6 M⊙ yr−1; 5σ). Taking into account the difference in sensitivity between the two surveys, we should have been able to detect ≥11.4 (≥7.6) of the 20 z ≃ 2 hosts – assuming that SFR evolves as ∝(1 + z)γ with γ= 2.5 (or γ= 0) respectively – whereas we found only four galaxies. Interestingly, all the z = 2 detected hosts have observed SFRs ≳ 9 M⊙ yr−1, well above our sensitivity limit, while at z = 1 they all have SFR < 9 M⊙ yr−1, an evolution that is in good agreement with the evolution of the SFR main sequence, i.e. with γ= 2.5. Moreover, we show that the z = 2 undetected hosts are not hidden under the quasar continuum after stacking our data. They also cannot be outside our surveyed area as this latter option runs against our sample selection criteria ( Å) and the known Wλ2796r–impact parameter relation for low-ionization ions. Hence, strong Mg ii absorbers could trace star-formation-driven winds in low-mass haloes (Mh≤ 1010.6 M⊙), provided that the winds do not extend beyond 20 kpc in order not to violate the evolution of the absorber number density dN/dz (Mg ii). Alternatively, our results imply that z = 2 galaxies traced by strong Mg ii absorbers do not form stars at a rate expected (3–10 M⊙ yr−1) for their (halo or stellar) masses, supporting the existence of a transition in accretion efficiency at Mh≃ 1011 M⊙. This scenario can explain both the detections and the non-detections.