We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on ${\mathbb{Z}}^d$, and its jump rate at $(\mathbf{x},t)$ is given by a fixed function φ of the state of a birth-and-death (BD) process at x, at time t; BD processes at different sites are independent and identically distributed, and φ is assumed non-increasing and vanishing at infinity. We derive a LLN and a CLT for the particle position when the environment is “strongly ergodic”. In the absence of a viable uniform lower bound for the jump rate, we resort instead to stochastic domination, as well as to a subadditive argument to control the time spent by the particle to perform n consecutive jumps; and we also impose conditions on the initial (product) environmental distribution.