Bone marrow stromal cells (BMSCs), whose populations contain multipotent skeletal stem cells with relevant therapeutic applications, are known to produce very heterogeneous colonies upon in vitro culture, a trait that may severely hinder the clinical usefulness of BMSC-based therapies. Therefore, reaching a better insight on the nature of such heterogeneity, as well as on the factors determining it, is important. Here, by using time-lapse microscopy, we study the structure of N=28 human BMSC colonies from six donors, each colony derived from a single cell, and trace their lineage trees up to the seventh generation. We confirm the presence of very significant inter-colony and intra-colony heterogeneities, both in the topology of the lineages and in the replicative kinetics of the colonies. We also find that topology and kinetics are strongly correlated, consistent with the existence of regulating factors linking the sub-population of inactive cells, which uniquely determine a lineage\'s topology, and that of active cells, which are the sole responsible for the proliferation rate of the colony. Finally, we submit each colony to an entropy-based inheritance test, which measures the degree of non-random clustering of inactive cells within the same branches of the lineage, and find a clear signature of hereditary transmission of the probability of emergence of inactive cells in the largest majority of the experimental lineages.