Neutral dynamics occur in evolution if all types are `effectively equal' in their reproductive success. Population dynamics with selection imply that the definition of `effectively equal' depends on the population size and the details of mutations. Genetic data for extremely large clonal populations indicates that many genes evolve neutrally, which current models can only explain if selection on those genes is completely absent. Such models typically consider the case where mutations are rare, so that population dynamics occurs at a different timescale to evolution and there are at most two competing types. However, if the probability of a mutation in the population as a whole is large, then the whole distribution of types must be considered. We show that this has important consequences for the occurrence of neutral dynamics. In highly connected type spaces, neutral dynamics can occur for all population sizes despite significant selective differences, via the forming of effectively neutral networks connecting rare neutral types. Biological implications include an explanation for the high diversity of rare types that survive in large clonal populations, and a theoretical justification for the use of neutral null models.