The interaction of fields with condensed matter during phase transitions produces a rich variety of physical phenomena. Self-assembly of liquid crystalline block copolymers (LC BCPs) in the presence of a magnetic field, for example, can result in highly oriented microstructures due to the LC BCP's anisotropic magnetic susceptibility. We show that such oriented mesophases can be produced using low-intensity fields (<0.5 T) that are accessible using permanent magnets, in contrast to the high fields (>4 T) and superconducting magnets required to date. Low-intensity field alignment is enabled by the addition of labile mesogens that coassemble with the system's nematic and smectic A mesophases. The alignment saturation field strength and alignment kinetics have pronounced dependences on the free mesogen concentration. Highly aligned states with orientation distribution coefficients close to unity were obtained at fields as small as 0.2 T. This remarkable field response originates in an enhancement of alignment kinetics due to a reduction in viscosity, and increased magnetostatic energy due to increases in grain size, in the presence of labile mesogens. These developments provide routes for controlling structural order in BCPs, including the possibility of producing nontrivial textures and patterns of alignment by locally screening fields using magnetic nanoparticles.