We have obtained the zero-temperature phase diagram of the kagomé antiferromagnet with Dzyaloshinskii-Moriya interactions in Schwinger-boson mean-field theory. We find quantum phase transitions (first or second order) between different topological spin-liquid and Néel-ordered phases (either the √3×√3 state or the so-called Q=0 state). In the regime of small Schwinger-boson density, the results bear some resemblances with exact diagonalization results and we briefly discuss some issues of the mean-field treatment. We calculate the equal-time structure factor (and its angular average to allow for a direct comparison with experiments on powder samples), which extends earlier work on the classical kagomé to the quantum regime. We also discuss the dynamical structure factors of the topological spin-liquid and the Néel-ordered phases.