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We develop a cross-border market model for two countries based on a continuous trading mechanism, in which the transmission capacities that enable transactions between market participants from different countries are limited. Our market model can be described by a regime-switching process alternating between active and inactive regimes, in which cross-border trading is possible respectively prohibited. Starting from a reduced-form representation of the two national limit order books, we derive a high-frequency approximation of the microscopic model, assuming that the size of an individual order converges to zero while the order arrival rate tends to infinity. If transmission capacities are available, the limiting dynamics are as follows: the queue size processes at the top of the two limit order books follow a four-dimensional linear Brownian motion in the positive orthant with oblique reflection at the axes. Each time the two best ask queues or the two best bid queues simultaneously hit zero, the queue size process is reinitialized. The capacity process can be described as a linear combination of local times and is hence of finite variation. The analytic tractability of the limiting dynamics allows us to compute key quantities of interest.