We propose a change in focus from the prevalent paradigm based on the branching property as a tool to analyze the structure of population models, to one based on the self-similarity property, which we also introduce for the first time in the setting of measure-valued processes. By extending the well-known Lamperti transformation into the infinite dimensional setting, we were able to embed and extend known results in population genetics within the self-similarity framework: we describe the frequency process of a larger class of measure-valued SS populations in terms of general Lambda Fleming-Viot processes. Our results demonstrate the potential power of the self-similar perspective for the study of populations whose total size varies stochastically over time, and in which the reproduction dynamics of the individuals are not independent from one another but are modulated by the total size of the population, allowing for more complex and realistic models. We also uncover a new duality relation between measure-valued processes and Lambda-coalescents which extends the well-known duality relation between Lambda Fleming-Viot processes and Lambda coalescents.