Vacancy formation and migration control self-diffusion in pure crystalline materials, whereas irradiation produces high concentrations of vacancy and self-interstitial atom defects, exceeding by many orders of magnitude the thermal equilibrium concentrations. The defects themselves, and the extended dislocation microstructure formed under irradiation, generate strongly spatially fluctuating strain and stress fields. These fields alter the local formation and migration enthalpies of defects, and give rise to the anisotropy of diffusion even if in the absence of stress the diffusion tensor is isotropic. We have performed textit{ab initio} calculations of formation and migration energies of vacancies in all the commonly occurring body-centred-cubic (BCC) metals, including alkaline, alkaline-earth and transition metals, and computed elastic dipole and relaxation volume tensors of vacancies at the equilibrium lattice positions and along the vacancy migration pathways. We find that in all the BCC metals the dipole tensor of a migrating vacancy at a saddle point exhibits an anti-crowdion character. Applied external stresses or the local stresses generated by dislocations may enhance or suppress anisotropic diffusion by altering the energy barriers with respect to the direction of migration of a defect.