The objectives of the present study are to investigate the unsteady two-dimensional laminar flow of a viscous incompressible electrically conducting polar fluid via a porous medium past a semi-infinite vertical porous moving plate in the presence of a transverse magnetic field. The plate moves with a constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially increasing or decreasing small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface which absorbs the polar fluid with a suction velocity varying with time. The effects of material parameters on the velocity and temperature fields across the boundary layer are investigated. The method of solution can be applied for small perturbation approximation. Numerical results of velocity distribution of polar fluids are compared with the corresponding flow problems for a Newtonian fluid. For a constant plate moving velocity with the given magnetic and permeability parameters, and Prandtl and Grashof numbers, the effect of increasing values of suction velocity parameter results in an increasing surface skin friction. It is also observed that the surface skin friction decreases by increasing the plate moving velocity.