We have probed the low temperature spin dynamics in two pyrochlore lattice compounds R 2Ti 2O 7, where the rare earth (R 3+) form a sublattice of tetrahedra linked by their corners such that geometrically derived magnetic frustration is possible. In Yb 2Ti 2O 7 (Yb 3+: S′=1/2, XY anisotropy), we show that below 0.24 K , the temperature of the known specific heat λ transition, there is no long range order. The transition, in fact, corresponds to a first-order change in the fluctuation rate of the Yb 3+ spins. Above the transition temperature, the rate is in the GHz range and it follows a thermal excitation law, whereas below the transition, the rate is in the MHz range and it is temperature independent. In Gd 2Ti 2O 7, where the S-state Gd 3+ ions are antiferromagnetically coupled, there is a magnetic phase transition near 1 K . As the temperature is lowered through the critical region, λ Z , the spin-lattice relaxation increases. On lowering the temperature to 21 mK , λ Z tends to a temperature independent value. Concomitant with the persisting electronic spin fluctuations, we observe oscillating components evidencing long-range spin correlations. Therefore, temperature independent spin dynamics is observed even after passing through a magnetic phase transition and even when the magnetic correlations are long range.