  We analyze Landauer's principle for repeated interaction systems consisting
of a reference quantum system $\mathcal{S}$ in contact with an environment
$\mathcal{E}$ which is a chain of independent quantum probes. The system
$\mathcal{S}$ interacts with each probe sequentially, for a given duration, and
the Landauer principle relates the energy variation of $\mathcal{E}$ and the
decrease of entropy of $\mathcal{S}$ by the entropy production of the dynamical
process. We consider refinements of the Landauer bound at the level of the full
statistics (FS) associated to a two-time measurement protocol of, essentially,
the energy of $\mathcal{E}$. The emphasis is put on the adiabatic regime where
the environment, consisting of $T \gg 1$ probes, displays variations of order
$T^{-1}$ between the successive probes, and the measurements take place
initially and after $T$ interactions. We prove a large deviation principle and
a central limit theorem as $T \to \infty$ for the classical random variable
describing the entropy production of the process, with respect to the FS
measure. In a special case, related to a detailed balance condition, we obtain
an explicit limiting distribution of this random variable without rescaling. At
the technical level, we obtain a non-unitary adiabatic theorem generalizing
that of [Commun. Math. Phys. (2017) 349: 285] and analyze the spectrum of
complex deformations of families of irreducible completely positive
trace-preserving maps.
