In this article, we propose a new set of multivariate stochastic models that capture time-varying seasonality within the vector innovations structural time-series (VISTS) framework. These models encapsulate exponential smoothing methods in a multivariate setting. The models considered are the local level, local trend, and damped trend VISTS models with an additive multivariate seasonal component. We evaluate the forecasting accuracy of these models against the forecasting accuracy of univariate alternatives using international tourist arrivals from 11 source countries to Australia and New Zealand. In general, the newly proposed multivariate models improve on forecast accuracy over the univariate alternatives.