A theoretical description is given for nonlinear formation mechanisms and properties of the polarization echoes in piezoelectric powders, earlier experimentally investigated in both the radiofrequency (rf) and the microwave (mw) frequency domains. For rf echoes, a phenomenological model is elaborated for dynamics of dislocations in mechanically vibrating piezoelectric particles resonantly excited by short-duration pulses of rf electric field. The model includes both the reversible and irreversible moving groups of dislocations generated by Frank - Read sources. The amplitude-dependent frequency change and amplitude-dependent damping obtained by use of this model constitute two of the three nonlinear mechanisms responsible for the formation of the polarization echoes in piezoelectric powders with the signals naturally consisting of both the dynamic and the memory components. As a third type of nonlinearity, the field - mode interaction, we take the nonlinear electrostriction. For mw echoes, it is proposed that the lack of memory components in the echoes is a consequence of absence of mobile dislocations in the powder material used. We suggest a somewhat modified form of the nonlinear mechanisms related to the pure lattice anharmonicity: amplitude-dependent dispersion and damping. General expressions for two-pulse rf echoes and mw echoes are derived by using together all three types of nonlinear mechanism inherent in each frequency domain. The numerical analysis of these expressions as a function of the pulse amplitudes, the pulse widths and the pulse separation shows good agreement between the theory and the existing experiments in a broad range of amplitudes and widths of the pulses. As a result, several important material constants relevant to the nonlinear mechanisms in (rf domain) and in ZnO (mw domain) are estimated.