A theoretical investigation of the dust ion-acoustic solitary, shock, and periodic waves has been made in a magnetised, dissipative, dusty plasma system. The system consists of cold ions, stationary dust grains, and non-inertial superthermal electrons and positrons. The Korteweg-de Vries-Burgers' (K-dVB) equation has been obtained by employing the reductive perturbation method (RPM). Using the appropriate travelling wave transformation, the model equation is transformed into a dynamical system. The different kinds of existing wave solutions are demonstrated in phase plots and time series diagrams based on appropriate parametric regions. The effect of superthermal electrons and positrons enhances the amplitude of both solitary and shock waves. With the variation of kinematic viscosity of ions, we observe the variation in breadth of the shock profile without affecting the amplitude. The alterations of the periodic wave solution with the involved system parameters are also shown in diagrammatic representations. The output of this present work could be useful to elucidate the bifurcation behaviour of solitary, shock, and periodic waves in an assortment of magnetised dissipative dusty plasma systems.