We investigate block-copolymer (BCP) thin film ordering kinetics during annealing across a moving in-plane temperature gradient. We operate in the so-called cold zone annealing (CZA) regime, where ordering temperatures are above the glass-transition, but well below the order-disorder transition. By measuring the order through the in-plane gradient, using atomic force microscopy and grazing-incidence small-angle X-ray scattering (GISAXS), we confirm that CZA greatly enhances ordering kinetics, as compared to uniform oven annealing. The maximal ordering occurs over a narrow range of the heating-up phase, and not during the subsequent cooling phase. The large grain sizes obtained using CZA are due to enhanced kinetics, and not the preferential formation of certain grain orientations. Kinetic enhancement is apparent even below the bulk glass-transition temperature. We suggest that the in-plane temperature gradient drives enhanced kinetics.