Broad-ridge laser diodes in the AlGaInN material system are efficient, high power laser sources in the visible spectral range. The broad waveguide gives rise to multiple active lateral modes, which lead to multiple longitudinal mode combs in the spectrum. In this work, the static longitudinal-lateral mode configuration as well as the mode dynamics are investigated, using different experimental techniques and numerical simulation. The static behavior is analyzed by high-resolution spectroscopy in near field and far field conditions. This allows to identify the active lateral modes, to distinguish lateral multi-mode operation from filamentation, and to study the influence of mode clustering and the current. With a streak camera setup, the lateral-spectral-temporal behavior is studied shortly after the pulse onset, as well as in the regime of mode competition. To investigate the mode competition process involving multiple lateral modes, a rate equation model is developed to numerically simulate the longitudinal-lateral mode dynamics. Based on this simulation approach, the influence of the most relevant parameters and the effect of mode clustering on the periodicity of the mode competition process is analyzed.