  This is the third in a series of papers in which we consider one-dimensional
Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by
starting from one-dimensional Random Walk in Random Environment (RWRE) and
resampling the environment along a sequence of deterministic times, called
refreshing times. In the present paper we explore two questions for general
refreshing times. First, we investigate how the recurrence versus transience
criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations
for RWCRE when RWRE is either recurrent or satisfies a classical central limit
theorem. We show that the answer depends in a delicate way on the choice of the
refreshing times. An overarching goal of our paper is to investigate how the
behaviour of a random process with a rich correlation structure can be affected
by resettings.
