We consider a stochastic individual-based model for the evolution of a population, whose space of possible traits is given by the vertices of a finite graph. The dynamics is driven by births, deaths, competition, and mutations along the edges of the graph. 
We are interested in the large population limit under a mutation rate given by a negative power of the carrying capacity K of the system. This results in several mutant traits being present at the same time and competing for invading the resident population.
We describe the time evolution of the orders of magnitude of each sub-population on the log K time scale, as K tends to infinity. 
Using techniques developed in [Champagnat, Méléard, Tran, 2019], we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. 
I will illustrate the theorem by  examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations. 
If time permits I will finish with an application to the phenomenon of evolutionary rescue.