We report on the spin dynamics of the strong spin-orbit coupled antiferromagnets Ba3Ru1−xIr x Ti2O9 (x = 0.5 and 0.8), which comprise a mixture of edge- and corner-sharing triangles. Muon spin-relaxation measurements give no hints of long-range magnetic order down to 25 mK. Rather, the muon spin-relaxation rates λ(T) show persistent spin dynamics below 1 K, indicating that fast fluctuations are dominant in spite of Ir4+(Jeff = 1/2)/Ru4+(S = 1) randomness. The muon spin depolarization of both compounds is well described by a stretched exponential function with the stretching exponent β = 0.4 (0.6) for x = 0.5 (0.8) at low temperatures, which is larger than β = 1/3 expected for a spin glass. Our results suggest that randomness in the spin number and the exchange interaction induces a partial spin freezing, but the majority of spins remain dynamically fluctuating.