Relativistic and non-relativistic particle acceleration along and across a magnetic field, and the generation of an electric field transverse to the magnetic field, both induced by almost perpendicularly propagating electrostatic waves in a relativistic magnetized plasma, are investigated theoretically on the basis of relativistic quasilinear transport equations. The electrostatic waves accelerate particles via Landau or cyclotron damping, and the ratio of parallel and perpendicular drift velocities vs||/vd can be proved to be proportional to k||/k⊥. Simultaneously, an intense cross-field electric field E0 = B0 × vd/c is generated via the dynamo effect owing to perpendicular particle drift to satisfy the generalized Ohm's law, which means that this cross-field particle drift is identical to E × B drift. The relativistic quasilinear transport equations for relativistic cross-field particle acceleration are derived by Lorentz transformation of the relativistic quasilinear momentum-space diffusion equation in the moving frame of reference without the electric field and the cross-field particle drift. They can be applied to the investigation of the relativistic perpendicular particle acceleration that may possibly occur in space plasmas.