Within the framework of nonrelativistic-QCD factorization, we calculate both the next-to-leading-order relativistic and QCD corrections to prompt pair production, with feed-down from mesons, via photon-photon collisions at future colliders, including the Future Circular Lepton Collider (FCC-ee), the Circular Electron Positron Collider (CEPC), and the Compact Linear Collider (CLIC). We present total cross sections and distributions in single transverse momentum and rapidity, as well as in pair invariant mass. The relativistic and QCD corrections both turn out to be large and negative. However, the production rates are large enough for useful experimental studies.