We present a simple and generic theoretical description of actin-based motility, where polymerization of filaments maintains propulsion. The dynamics is driven by polymerization kinetics at the filaments' free ends, crosslinking of the actin network, attachment and detachment of filaments to the obstacle interface and entropic forces. We show that spontaneous oscillations in the velocity emerge due to a push-pull mechanism in a broad range of parameter values, and compare our findings with experiments.