A core challenge for modern biology is how to infer the trajectories of individual cells from population-level time courses of high-dimensional gene expression data. Birth and death of cells present a particular difficulty: existing trajectory inference methods cannot distinguish variability in net proliferation from cell differentiation dynamics, and hence require accurate prior knowledge of the proliferation rate. In this talk, I will present first Global Waddington-OT (gWOT), a method based on regularized optimal transport which performs trajectory inference with rigorous theoretical guarantees when birth and death can be neglected, or are known prior to the observation. I will then show how recent CRISPR-based measurement technologies, by giving access to the lineage tree describing shared ancestry within a population of cells, allow to build on gWOT to disentangle proliferation and differentiation without any prior knowledge. Death and/or subsampling may nevertheless introduce a bias in the inferred trajectory, that we describe explicitly and argue to be inherent to these lineage tracing data.