We propose a simple scaling theory describing the variation of the mean first passage time (MFPT) Y(N, M) of a regular block copolymer of chain length N and block size M which is dragged through a selective liquid-liquid interface by an external field B. The theory predicts a non-Arrhenian Y vs B relationship which depends strongly on the size of the blocks, M, and rather weakly on the total polymer length, N. The overall behavior is strongly influenced by the degree of selectivity between the two solvents O. The variation of Y(N, M) with N and M in the regimes of weak and strong selectivity of the interface is also studied by means of computer simulations using a dynamic Monte Carlo coarse-grained model. Good qualitative agreement with theoretical predictions is found. The MFPT distribution is found to be well described by a Gamma-distribution. Transition dynamics of ring and telechelic polymers is also examined and compared to that of the linear chains. The strong sensitivity of the "capture" time Y(N, M) with respect to block length M suggests a possible application as a new type of chromatography designed to separate and purify complex mixtures with different block sizes of the individual macromolecules.