Spatially resolved transcriptomics (SRT) measures mRNA transcripts at thousands of locations within a tissue slice, revealing spatial variations in gene expression and cell types. SRT has been applied to tissue slices from multiple time points during the development of an organism. We introduce developmental spatiotemporal optimal transport (DeST-OT), a method to align spatiotemporal transcriptomics data using optimal transport (OT). DeST-OT uses semi-relaxed OT to model cellular growth, death, and differentiation processes. We also derive a growth distortion metric and a migration metric to quantify the plausibility of spatiotemporal alignments. DeST-OT outperforms existing methods on the alignment of spatiotemporal transcriptomics data from developing mouse kidney and axolotl brain. DeST-OT estimated growth rates also provide insights into the gene expression programs governing the growth and differentiation of cells over space and time.