Recent studies of in vitro evolution of DNA via protein binding indicate that the evolution behavior is qualitatively different in different parameter regimes. I here present a general theory that is valid for a wide range of parameters, and which reproduces and extends previous results. Specifically, the mean-field theory of a general translation-invariant model can be reduced to the basic diffusion equation with a dynamic boundary condition. The simple analytical form yields both quantitatively accurate predictions and valuable insight into the principles involved. In particular, I introduce a cutoff criterion for finite populations that illustrates both of these qualities.