We numerically explore synthetic crystal diamond for realizing novel light sources in ranges which are up to now difficult to achieve with other materials, such as sub-10-fs pulse durations and challenging spectral ranges. We assess the performance of on-chip diamond waveguides for controlling light generation by means of nonlinear soliton dynamics. The considered silica-embedded diamond waveguide model exhibits two zero-dispersion points, delimiting an anomalous dispersion range that exceeds an octave. Various propagation dynamics, including supercontinuum generation by soliton fission, can be realized in diamond photonics. In contrast to usual silica-based optical fibers, where such processes occur on the scale of meters, in diamond millimeter-scale propagation distances are sufficient. Unperturbed soliton-dynamics prior to soliton fission allow identifying a pulse self-compression scenario that promises record-breaking compression factors on chip-size propagation lengths.