Transport-based density estimation methods are gaining popularity due to their efficiency in generating samples from the target density to be approximated. In this talk, we introduce a sequential framework for constructing a deterministic transport map as the composition of Knothe-Rosenblatt (KR) maps built in a greedy manner. The key ingredient is the introduction of an arbitrary sequence of 'bridging densities,' which is used to guide the sequential algorithm. While tempered (or annealed) bridging densities are natural to use in the context of Bayesian inverse problems, diffusion-based bridging densities are more suitable when the target density is known from samples only. To build each of the KR maps, we first estimate the intermediate density using Sum-of-Squares (SoS) density surrogates, and then we analytically extract the KR map of that precomputed approximation. We also propose a convergence analysis of the resulting algorithm with respect to the alpha-divergence, which generalizes previous results from the literature. Additionally, we numerically demonstrate our method on several benchmarks, including Bayesian inference problems and unsupervised learning tasks.