Seemingly minor details of mathematical and computational models of evolution are known to change the effect of population structure on the outcome of evolutionary processes. For example, birth-death dynamics often result in amplification of selection, while death-birth processes have been associated with suppression. In many biological populations the interaction structure is not static. Instead, members of the population are in motion and can interact with different individuals at different times. In this work we study populations embedded in a flowing medium; the interaction network is then time dependent. We use computer simulations to investigate how this dynamic structure affects the success of invading mutants, and compare these effects for different coupled birth and death processes. Specifically, we show how the speed of the motion impacts the fixation probability of an invading mutant. Flows of different speeds interpolate between evolutionary dynamics on fixed heterogeneous graphs and well-stirred populations; this allows us to systematically compare against known results for static structured populations. We find that motion has an active role in amplifying or suppressing selection by fragmenting and reconnecting the interaction graph. While increasing flow speeds suppress selection for most evolutionary models, we identify characteristic responses to flow for the different update rules we test. In particular we find that selection can be maximally enhanced or suppressed at intermediate flow speeds.