We investigate field-driven domain-wall dynamics in thin ferromagnetic layers in the precessional regime using an analytical model that goes beyond the rigid wall approximation and takes into account the flexing modes of domain walls. This model allows us to turn on and off the stray field from domains adjacent to the domain wall in order to elucidate its role. We determine the eigenfrequencies of the flexing modes. The domain-wall flexing instabilities are shown to exist even without the stray field. The amplitude of the flexing modes shows a maximum when the magnetization precession frequency is equal to their eigenfrequency. At maximum amplitude of the flexing modes, the domain-wall velocity exhibits bumps even without the stray field. The stray field further enhances the velocity and broadens the bumps. This study is completed by micromagnetic simulations that show excellent agreement with the analytical model. The role of the various spatial modes in the velocity enhancement is clarified. Additional fractional frequencies appear in the time-dependent amplitude of the flexing modes, in analogy with the behavior of parametric oscillators.