Populations of physiologically vital mitochondrial DNA (mtDNA) molecules evolve in cells under control from the nucleus. The evolution of populations of mixed mtDNA types is complicated and poorly understood, and variability of these controlled admixtures plays a central role in the inheritance and onset of genetic disease. Here, we develop a mathematical theory describing the evolution and variability in these stochastic populations for any type of cellular control, showing that cell-to-cell variability in mtDNA, and mutant load, inevitably increases with time, according to rates which we derive and which are notably independent of the mechanistic details of feedback signalling. We show with a set of experimental case studies that this theory explains disparate quantitative results from classical and modern experimental and computational studies on heteroplasmy variance in different species. We demonstrate that our general model provides a host of specific insights, including a modification of the often-used but hard-to-interpret Wright formula to correspond directly to biological observables, the ability to quantify selective and mutational pressure in mtDNA populations, and the pronounced variability inevitably arising from the action of possible mtDNA quality-control mechanisms. Our general theoretical framework, supported by existing experimental results, thus helps understand and predict the evolution of stochastic mtDNA populations in cell biology.