Microbial populations often have complex spatial structures, with homogeneous competition holding only at a local scale. Population structure can strongly impact evolution, in particular by affecting the fixation probability of mutants. Here, we propose a model of structured microbial populations on graphs, where each node of the graph contains a well-mixed deme whose size can fluctuate, and where migrations are independent from birth and death events. We study analytically and numerically the mutant fixation probabilities in different structures, in the rare migration regime. In particular, we demonstrate that the star graph continuously transitions between amplifying and suppressing natural selection as migration rate asymmetry is varied. This elucidates the apparent paradox in existing constant-size models on graphs, where the star is an amplifier or a suppressor depending on the details of the dynamics or update rule chosen, e.g. whether each birth event precedes or follows a death event. The celebrated amplification property of the star graph for large populations is preserved in our model, for specific migration asymmetry. We further demonstrate a general mapping between our model and constant-size models on graphs, under a constraint on migration rates, which directly stems from assuming constant size. By lifting this constraint, our model reconciles and generalizes previous results, showing that migration rate asymmetry is key to determining whether a given population structure amplifies or suppresses natural selection.SummaryA key question in evolution is how likely a mutant is to take over. This depends on natural selection and on stochastic fluctuations. Compared to a well-mixed population, some spatial structures have been predicted to amplify or suppress the impact of natural selection. This can have a major impact on evolution, as well as applications in biotechnology, e.g. by facilitating selection in directed evolution of biomolecules. We introduce a model for complex structured populations that generalizes over existing ones by allowing population size to fluctuate. We demonstrate that whether a given population structure amplifies or suppresses selection strongly depends on the asymmetry of migration rates. Our results are universal in the sense that they do not depend on update rules.