The mechanisms that generate dynamic spatial patterns within proliferating tissues are poorly understood, largely because of difficulties in unravelling interactions between cell specification, polarity, asymmetric division, rearrangements, and growth. We address this problem for stomatal spacing in plants, which offer the simplifying advantage that cells do not rearrange. By tracking lineages and gene activities over extended periods, we show that limited stem cell behavior of stomatal precursors depends on maintenance of the SPEECHLESS (SPCH) transcription factor in single daughter cells. Modeling shows how this property can lead to observed stereotypical stomata lineages through a postmitotic polarity-switching mechanism. The model predicts the location of a polarity determinant BASL over multiple divisions, which we validate experimentally. Our results highlight the dynamic two-way interactions between stem cells and their neighborhood during developmental patterning.