Vortex dynamics is studied in coupled multilayer structures, containing N 24-Å-thick DyBa₂Cu₃O₇ layers, each separated from the next by 96 Å of (Y₁₋ₓPrₓ)Ba₂Cu₃O₇7 (x=0.4, 0.55). When the magnetic field is parallel to the c axis, we find that the activation energy U for flux motion increases linearly with the number of superconducting layers N in the structure for N<3. This linear increase is the result of the coupled motion of pancake vortices belonging to different DyBa₂Cu₃O₇ layers. For larger N samples the activation energy saturates meaning that the vortex lattice is turning three dimensional. In contrast to samples whose vortex lattice is purely two dimensional and for which we find the activation energy for flux motion proportional to log₁₀B, samples in these series show a crossover to a power-law behavior, U≈B⁻⁰˙⁵, at a magnetic field H* which decreases as N increases.