The lattice dynamics of Pb-containing perovskite oxides are investigated theoretically for the transition metal series Ti, Zr, Hf, in order to elucidate their commonalities and their distinctions. For all three compounds, pronounced precursor effects are found to their phase transition temperatures, which get more pronounced the heavier the central transition metal ion is. In addition, a competition between a polar and an antiferrodistortive instability is predicted to take place, which is strongly mass dependent. While in PbTiO3 the polar instability wins, both instabilities are active in PbZrO3, whereas in PbHfO3 the antiferrodistortive phase transition dominates the dynamics. For all three compounds, marked anomalies in the elastic constants are predicted, which are most pronounced in PbHfO3. Experimental results for elastic anomalies preceding the phase transition, which agree qualitatively with the model calculations are presented for PbHfO3.