Motivated by recent biological experiments, we emphasize the effects of small populations in various biological/medical contexts related to evolution such as invasion of mutant cells or emergence of antibiotic resistances. Our main mathematical challenge is to quantify such effects in particular on macroscopic approximations. In order to track individuals and to take into account small populations, we are led to stochastic multiscale models while the limiting macroscopic equations should involve non local non linear partial differential equations. Such approximations are used to analyse the conditions for a mutant invasion from a monotype population and to provide the mutant probability of fixation and its time to fixation. That allows to exhibit a rare mutation scale assumption yielding a timescale separation between competition and mutation. Under these assumptions, the stochastic measure-valued process at the mutation timescale converges to a jump process which describes the successive invasions of successful mutants. The gene transfer can drastically affect the evolutionary outcomes. For fast mutation timescales, preliminary numerical simulations indicate that these models should exhibit many surprising asymptotic behaviours such as cyclic behaviours. These phenomena are mathematically explored on a simple model. Population sizes and times are considered in a log scale to keep track of small subpopulations that have negligible sizes compared with the size of the resident population. Explicit criteria on the model parameters are given to characterise the possible evolutionary outcomes. An important ingredient for the proofs lies in comparison of the stochastic population process with linear or logistic birth-death processes with immigration. The impact of these time and size scales on macroscopic approximations is also investigated, leading to a new class of Hamilton-Jacobi equations with constraint boundary conditions.