The present study explores the influence of viscous dissipation, Joule heating, and double dispersion on unsteady, free convective magnetohydrodynamics (MHD) flow of an incompressible Casson fluid over a vertical cone and flat plate saturated with porous medium subject to variable viscosity and variable electrical conductivity. The governing coupled, nonlinear partial differential equations are solved by Crank–Nicolson method. The effects of various significant parameters on flow, heat, and mass transfer characteristics are displayed in the form of figures and tables. The results indicate that the presence of variable viscosity parameter meagerly accelerates the fluid flow. It is observed that heat transfer is enhanced for increasing the thermal dispersion parameter and Eckert number.