We develop two models for the temporal evolution of extreme events of multivariate k $$ k $$ th order Markov processes. The foundation of our methodology lies in the conditional extremes model of Heffernan and Tawn (Journal of the Royal Statistical Society: Series B (Methodology), 2014, 66, 497–546), and it naturally extends the work of Winter and Tawn (Journal of the Royal Statistical Society: Series C (Applied Statistics), 2016, 65, 345–365; Extremes, 2017, 20, 393–415) and Tendijck et al. (Environmetrics 2019, 30, e2541) to include multivariate random variables. We use cross‐validation‐type techniques to develop a model order selection procedure, and we test our models on two‐dimensional meteorological‐oceanographic data with directional covariates for a location in the northern North Sea. We conclude that the newly‐developed models perform better than the widely used historical matching methodology for these data.