Lasers capable of emitting two or more distinct wavelengths with a control over their power balance appear as promising and versatile key devices in the context of THz signal generation, telecommunication or wavelength conversion using optical injection. For such applications, fast wavelength switching is required but devices implying tunability through thermal or mechanical actions can be slow or bulky. Photonic integration has the potential to overcome these obstacles. While merging the beam of distinct lasers on the same chip seems to be a straightforward solution, the absence of modal competition and phase noise correlation limits their use in some applications.Here, we consider DBR-based multi-cavity lasers with multiple wavelengths sharing a common broadband gain medium. The wavelength switching principle relies on controlling the phase in a monolithically integrated optical feedback cavity. In this contribution, by monitoring the emitted time-resolved power signal, we characterize the dynamics of the wavelength switching mechanism and report a measured response time below 4 ns. Additionally, from numerical investigations using a multi-mode extension of the Lang-Kobayashi rate equations, we identify key parameters influencing the switching time.