We have recently reported in [PRL 110, 064103 (2013)] that in systems which otherwise do not show oscillatory dynamics, the interplay between pinning to a defect and pulling by drift allows the system to exhibit excitability and oscillations. Here, we build on this work and present a detailed bifurcation analysis of the various dynamical instabilities that result from the competition between a pulling force generated by the drift and a pinning of the solitons to spatial defects. We show that oscillatory and excitable dynamics of dissipative solitons find their origin in multiple codimension-two bifurcation points. Moreover, we demonstrate that these mechanisms leading to these dynamical regimes are generic for any system admitting dissipative solitons.