Multidimensional measures of poverty have become standard as complementary indicators of poverty in many countries. Multidimensional poverty calculations typically comprise three indices: the multidimensional headcount, the average deprivation share among the poor, and the adjusted headcount ratio. While several decomposition methodologies are available for the last index, less attention has been paid to decomposing the multidimensional headcount, despite the attention it receives from policy makers. This paper proposes an application of existing methodologies that decompose welfare aggregates--based on counterfactual simulations--to break up the changes of the multidimensional poverty headcount into the variation attributed to each of its dimensions. This paper examines the potential issues of using counterfactual simulations in this framework, proposes approaches to assess these issues in real applications, and suggests a methodology based on rank preservation within strata, which performs positively in simulations. The methodology is applied in the context of the recent reduction of multidimensional poverty in Colombia, finding that the dimensions associated with education and health are the main drivers behind the poverty decline.