The present study investigates the generalized dispersion of a solute in an incompressible MHD flow via a rectangular channel with injectable or suctioned walls. The mathematical model of dispersion suggests a distinct type of mass flux expressed as a fractional partial differential equation based on the time-fractional Caputo derivative. The mass flow in the model under investigation is determined by both the concentration gradient and its historical evolution. A constant external magnetic field is provided transverse to the flow direction. The analysis and discussion of the analytical solution for the advection velocity are performed in relation to the Hartmann number and the suction/injection Reynolds number. To determine the solute concentration in space and time, the unstable fractional convection–diffusion equation is analytically solved. The polynomial in the geographic variable y that has coefficients that depend on the spatial variable x and the time t is the analytical solution of the concentration. The effects of the fractional order of the Caputo derivative, Reynolds number, Hartmann number, and Peclet number on the advection–diffusion process are examined using numerical simulations of the analytical solution of the solute concentration.