Domain wall dynamics in a perpendicularly magnetized system is studied by means of micromagnetic simulations in which disorder is introduced as a dispersion of both the easy-axis orientation and the anisotropy constant over regions reproducing a granular structure of the material. High field dynamics show a linear velocity-field relationship and an additional grain size dependent velocity shift, weakly dependent on both applied field and intrinsic Gilbert's damping parameter. We find the origin of this velocity shift in the nonhomogeneous in-plane effective field generated by the tilting of anisotropy easy axis introduced by disorder. We show that a one-dimensional analytical approach cannot predict the observed velocities and we augment it with the additional dissipation of energy arising from internal domain wall dynamics triggered by disorder. This way we prove that the main cause of higher velocity is the ability of the domain wall to irradiate energy into the domains, acquired with a precise feature of disorder.