One-dimensional topological pumping of matter waves in two overlaid optical lattices moving with respect to each other is considered in the presence of attractive nonlinearity. It is shown that there exists a threshold nonlinearity level above which the matter transfer is completely arrested. Below this threshold the transfer of both dispersive wavepackets and solitons occurs in accordance with the predictions of the linear theory, i.e. it is quantized and determined by the dynamical Chern numbers of the lowest band. The breakdown of the transport is also explained by nontrivial topology of the bands. In that case, the nonlinearity induces Rabi oscillations of atoms between two (or more) lowest bands. If the sum of the dynamical Chern numbers of the populated bands is zero, the oscillatory dynamics of a matter soliton in space occurs, which corresponds to the transport breakdown. Otherwise the sum of the Chern numbers of the nonlinearity-excited bands determines the direction and magnitude of the average velocity of matter solitons that remains quantized and admits fractional values. Thus, even in strongly nonlinear regime the topology of the linear bands is responsible for the evolution of solitons. The transition between different dynamical regimes is accurately described by the perturbation theory for solitons.