We present and develop a general dispersive framework allowing us to construct representations of the amplitudes for the processes $P\pi\to\pi\pi$, $P=K, \eta$, valid at the two-loop level in the low-energy expansion. The construction proceeds through a two-step iteration, starting from the tree-level amplitudes and their S and P partial-wave projections. The one-loop amplitudes are obtained for all possible configurations of pion masses. The second iteration is performed in the cases where either all masses of charged and neutral pions are equal or for the decay into three neutral pions. Issues related to analyticity properties of the amplitudes and their lowest partial-wave projections are given particular attention.