Living tissues display fluctuations—random spatial and temporal variations of tissue properties around their reference values—at multiple scales. It is believed that such fluctuations may enable tissues to sense their state or their size. Recent theoretical studies developed specific models of fluctuations in growing tissues and predicted that fluctuations of growth show long-range correlations. Here, we elaborated upon these predictions and we tested them using experimental data. We first introduced a minimal model for the fluctuations of any quantity that has some level of temporal persistence or memory, such as concentration of a molecule, local growth rate, or mechanical property. We found that long-range correlations are generic, applying to any such quantity, and that growth couples temporal and spatial fluctuations, through a mechanism that we call “fluctuation stretching”—growth enlarges the length scale of variation of this quantity. We then analyzed growth data from sepals of the model plant Arabidopsis and we quantified spatial and temporal fluctuations of cell growth using the previously developed cellular Fourier transform. Growth appears to have long-range correlations. We compared different genotypes and growth conditions: mutants with lower or higher response to mechanical stress have lower temporal correlations and longer-range spatial correlations than wild-type plants. Finally, we used theoretical predictions to merge experimental data from all conditions and developmental stages into a unifying curve, validating the notion that temporal and spatial fluctuations are coupled by growth. Altogether, our work reveals kinematic constraints on spatiotemporal fluctuations that have an impact on the robustness of morphogenesis.