Flexible filaments moving in viscous fluids are ubiquitous in the natural microscopic world. For example, the swimming of bacteria and spermatozoa as well as important physiological functions at organ level, such as the cilia-induced motion of mucus in the lungs, or individual cell level, such as actin filaments or microtubules, all employ flexible filaments moving in viscous fluids. As a result of fluid-structure interactions, a variety of nonlinear phenomena may arise in the dynamics of such moving flexible filaments. In this paper we derive the mathematical tools required to study filament-driven propulsion in the asymptotic limit of stiff filaments. Motion in the rigid limit leads to hydrodynamic loads which deform the filament and impact the filament propulsion. We first derive the general mathematical formulation and then apply it to the case of a helical filament, a situation relevant for the swimming of flagellated bacteria and for the transport of artificial, magnetically actuated motors. We find that, as a result of flexibility, the helical filament is either stretched or compressed (conforming previous studies) and additionally its axis also bends, a result which we interpret physically. We then explore and interpret the dependence of the perturbed propulsion speed due to the deformation on the relevant dimensionless dynamic and geometric parameters.