  We consider random walks in dynamic random environments, with an environment
generated by the time-reversal of a Markov process from the oriented
percolation universality class. If the influence of the random medium on the
walk is small in space-time regions where the medium is typical, we obtain a
law of large numbers and an averaged central limit theorem for the walk via a
regeneration construction under suitable coarse-graining.
  Such random walks occur naturally as spatial embeddings of ancestral lineages
in spatial population models with local regulation. We verify that our
assumptions hold for logistic branching random walks when the population
density is sufficiently high.
