  We consider random walks in dynamic random environments given by Markovian
dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary
distribution $\mu$ and satisfies the Poincar\'e inequality w.r.t. $\mu$. The
random walk is a perturbation of another random walk (called "unperturbed"). We
assume that also the environment viewed from the unperturbed random walk has
stationary distribution $\mu$. Both perturbed and unperturbed random walks can
depend heavily on the environment and are not assumed to be finite-range. We
derive a law of large numbers, an averaged invariance principle for the
position of the walker and a series expansion for the asymptotic speed. We also
provide a condition for non-degeneracy of the diffusion, and describe in some
details equilibrium and convergence properties of the environment seen by the
walker. All these results are based on a more general perturbative analysis of
operators that we derive in the context of $L^2$-bounded perturbations of
Markov processes by means of the so-called Dyson-Phillips expansion.
