The localization kinetics of a regular block copolymer of total length N and block size M at a selective liquid-liquid interface is studied in the limit of strong segregation between hydrophobic and polar segments in the chain. We propose a simple analytic theory based on scaling arguments which describes the relaxation of the initial coil into a flat-shaped layer for the cases of both Rouse and Zimm dynamics. For Rouse dynamics, the characteristic times for attaining equilibrium values of the gyration radius components perpendicular and parallel to the interface are predicted to scale with block length M and chain length N as tau(perpendicular to) proportional to M1+2v (here v approximate to 0.6 is the Flory exponent) and as tau(parallel to) proportional to N-2, although initially the characteristic coil flattening time is predicted to scale with block size as proportional to M. Since typically N >> M for multiblock copolymers, our results suggest that the flattening dynamics proceeds faster perpendicular rather than parallel to the interface, in contrast to the case of Zimm dynamics where the two components relax with comparable rate, and proceed considerably slower than in the Rouse case. We also demonstrate that, in the case of Rouse dynamics, these scaling predictions agree well with the results of Monte Carlo simulations of the localization dynamics. A comparison to the localization dynamics of random copolymers is also carried out.