<p>The transient electrohydrodynamics of a coaxial liquid column subjected to a weak uniform electric field is examined analytically within the leaky-dielectric framework. Using four representative fluid systems—chosen to capture the distinct steady-state flow configurations that can arise in such multilayer arrangements—we study the evolution of the flow field and interface deformation toward the steady state. The flow evolution is interpreted through dividing streamlines, which clearly reveal the creation, migration, and disappearance of recirculating regions in the core, shell, and ambient; this diagnostic also provides a framework that can be extended to mixing and transport analyses in related settings. For small departures from circular geometry, the interface dynamics are governed by two characteristic relaxation times whose magnitudes depend on the kinetic (self-relaxation) and cross-coupling coefficients of the problem. Depending on the material contrasts, the two interfaces may both deform monotonically, or one may exhibit a non-monotonic evolution while the other remains monotonic. The ratios of electric conductivities and permittivities of the participating fluids play a key role in determining both the transient flow topology and the approach to the steady state. Comparison with the corresponding 3D spherical configuration, enabled by an asymptotic analysis, shows that 2D and 3D systems, although qualitatively similar, differ substantially in magnitude owing to their distinct dipolar and geometric scalings as well as curvature, and therefore cannot be used interchangeably. The accompanying asymptotic analysis unifies the behavior of the extreme conductivity-contrast regimes and yields simple, geometry-dependent 2D/3D relations for the steady capillary deformation; the same methodology extends naturally to other limiting regimes, including those involving strong permittivity or viscosity contrasts.</p>