Abstract This work is concerned with peristaltic flow as a result of the influence of ultrasonic radiation on the flow of a liquid through the gap between two coaxial tubes (annulus) which deforms the wall of the outer tube in the shape of travelling transversal waves. The problem studies the effect of slip velocity and viscoelasticity on the dynamics of a compressible and electrically conducting Jeffrey fluid through a porous medium in the presence of a constant magnetic field. Navier-Stokes equations for the annulus are solved by means of a perturbation analysis, in which the ratio of the wave amplitude to the outer tube radius is a small parameter. In the second order approximation, the net flow induced by the travelling wave is calculated for various values of Reynolds number, relaxation time, retardation time, magnetic parameter, slip parameter, permeability, and annulus radius ratio. The calculations disclose that the compressibility, the radius ratio, and the non-Newtonian effects have a strong influence on the net flow rate induced. This problem has numerous applications in various branches of science, including stimulation of fluid flow in annuli under the effect of elastic waves, the production process of oil, and studies of blood flow dynamics in living creatures (catheter in an artery). The present study investigates novelties brought about into the classic peristaltic mechanism by inclusion of the annulus radius ratio and the non-Newtonian effects that were found to lessen the oscillatory behavior of the net flow rate. Several results of other models can be deduced as the limiting cases of our situation. Finally, variations of the emerging parameters embedded in the flow system are discussed numerically and graphically.