As part of our on-going project on the homogeneous chemical characterisation of multiple stellar populations in globular clusters (GCs), we studied NGC 5634, associated to the Sagittarius dwarf spheroidal galaxy, using high-resolution spectroscopy of red giant stars collected with VLT/FLAMES. We present here the radial velocity distribution of the 45 observed stars, 43 of which are cluster members, the detailed chemical abundance of 22 species for the seven stars observed with UVES-FLAMES, and the abundance of six elements for stars observed with GIRAFFE. On our homogeneous UVES metallicity scale, we derived a low-metallicity Fe/H =−1.867 ± 0.019 ± 0.065 dex (±statistical ±systematic error) with σ = 0.050 dex (7 stars). We found the normal anticorrelations between light elements (Na and O, Mg and Al), a signature of multiple populations typical of massive and old GCs. We confirm the associations of NGC 5634 to the Sgr dSph, from which the cluster was lost a few Gyr ago, on the basis of its velocity and position, and the abundance ratios of α and neutron capture elements.