An experiment to search for double-β decay processes in 96Ru and 104Ru, which are accompanied by γ rays, has been realized in the underground Gran Sasso National Laboratories of the I.N.F.N. (Italy). Ruthenium samples with masses of ≈0.5–0.7 kg were measured with the help of ultralow-background high-purity Ge γ -ray spectrometry. After 2162 h of data taking the samples were deeply purified to reduce the internal contamination of 40K. The last part of the data has been accumulated over 5479 h. New improved half-life limits on 2β+/εβ+/2ε processes in 96Ru have been established on the level of 1020 yr, in particular for decays to the ground state of 96Mo: T 2ν2β+ 1/2 1.4 × 1020 yr, T 2νεβ+ 1/2 8.0 × 1019 yr, and T 0ν2K 1/2 1.0 × 1021 yr (all limits are at 90% C.L.). The resonant neutrinoless double-electron captures to the 2700.2 and 2712.7 keV excited states of 96Mo are restricted as T 0νKL 1/2 2.0 × 1020 yr and T 0ν2L 1/2 3.6 × 1020 yr, respectively. Various two-neutrino and neutrinoless 2β half-lives of 96Ru have been estimated in the framework of the quasiparticle random-phase approximation approach. In addition, the T1/2 limit for 0ν2β− transitions of 104Ru to the first excited state of 104Pd has been set as 6.5 × 1020 yr.