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Significance What defines the number and dynamics of the stem cells that generate and renew biological tissues? Although several molecular markers have been described to predict stem cell potential, we propose a complementary approach that mathematically describes “stemness” as an emergent property arising from a stochastic competition for space. We predict from that competition the robust emergence of a region made of functional stem cells, as well as give simple predictions on lineage-survival probability. We test our results with data obtained from intravital live-imaging experiments in mammary gland development, existing data from kidney development, and from the self-renewal of the crypt to show that our framework can predict the number of functional stem cells and lineage-survival probability., Understanding to what extent stem cell potential is a cell-intrinsic property or an emergent behavior coming from global tissue dynamics and geometry is a key outstanding question of systems and stem cell biology. Here, we propose a theory of stem cell dynamics as a stochastic competition for access to a spatially localized niche, giving rise to a stochastic conveyor-belt model. Cell divisions produce a steady cellular stream which advects cells away from the niche, while random rearrangements enable cells away from the niche to be favorably repositioned. Importantly, even when assuming that all cells in a tissue are molecularly equivalent, we predict a common (“universal”) functional dependence of the long-term clonal survival probability on distance from the niche, as well as the emergence of a well-defined number of functional stem cells, dependent only on the rate of random movements vs. mitosis-driven advection. We test the predictions of this theory on datasets of pubertal mammary gland tips and embryonic kidney tips, as well as homeostatic intestinal crypts. Importantly, we find good agreement for the predicted functional dependency of the competition as a function of position, and thus functional stem cell number in each organ. This argues for a key role of positional fluctuations in dictating stem cell number and dynamics, and we discuss the applicability of this theory to other settings.