In this article we study a class of self-interacting Markov chain models. We propose a novel theoretical basis based on measure-valued processes and semigroup techniques to analyze its asymptotic behavior as the time parameter tends to infinity. We exhibit different types of decays to equilibrium, depending on the level of interaction. We illustrate these results in a variety of examples, including Gaussian or Poisson self-interacting models. We analyze the long-time behavior of a new class of evolutionary self-interacting chain models. These genetic type algorithms can also be regarded as reinforced stochastic explorations of an environment with obstacles related to a potential function.