Hematopoietic Stem Cells (HSC) are the cells that give rise to all other blood cells and, as such, they are crucial in the healthy development of individuals. Wiskott-Aldrich Syndrome (WAS) is a severe disorder affecting the regulation of hematopoietic cells and is caused by mutations in the WASP gene. We consider data from a revolutionary gene therapy clinical trial, where HSC harvested from 3 WAS patients’ bone marrow have been edited and corrected using viral vectors. Upon re-infusion into the patient, the HSC multiply and differentiate into other cell types. The aim is to unravel the cell multiplication and cell differentiation process, which has until now remained elusive. This paper models the replenishment of blood lineages resulting from corrected HSC via a multivariate, density-dependent Markov process and develops an inferential procedure to estimate the dynamic parameters given a set of temporally sparsely observed trajectories. Starting from the master equation, we derive a system of non-linear differential equations for the evolution of the first- and second-order moments over time. We use these moment equations in a generalized method-of-moments framework to perform inference. The performance of our proposal has been evaluated by considering different sampling scenarios and measurement errors of various strengths using a simulation study. We also compared it to another state-of-the-art approach and found that our method is statistically more efficient. By applying our method to the Wiskott-Aldrich Syndrome gene therapy data we found strong evidence for a myeloid-based developmental pathway of hematopoietic cells where fates of lymphoid and myeloid cells remain coupled even after the loss of erythroid potential. All code used in this manuscript can be found in the online Supplement, and the latest version of the code is available at github. com/dp3ll1n/SLCDP_v1.0.