  We study Landauer's Principle for Repeated Interaction Systems (RIS)
consisting of a reference quantum system $\mathcal{S}$ in contact with a
structured environment $\mathcal{E}$ made of a chain of independent quantum
probes; $\mathcal{S}$ interacts with each probe, for a fixed duration, in
sequence. We first adapt Landauer's lower bound, which relates the energy
variation of the environment $\mathcal{E}$ to a decrease of entropy of the
system $\mathcal{S}$ during the evolution, to the peculiar discrete time
dynamics of RIS. Then we consider RIS with a structured environment
$\mathcal{E}$ displaying small variations of order $T^{-1}$ between the
successive probes encountered by $\mathcal{S}$, after $n\simeq T$ interactions,
in keeping with adiabatic scaling. We establish a discrete time non-unitary
adiabatic theorem to approximate the reduced dynamics of $\mathcal{S}$ in this
regime, in order to tackle the adiabatic limit of Landauer's bound. We find
that saturation of Landauer's bound is equivalent to a detailed balance
condition on the repeated interaction system, reflecting the non-equilibrium
nature of the repeated interaction system dynamics. This is to be contrasted
with the generic saturation of Landauer's bound known to hold for continuous
time evolution of an open quantum system interacting with a single thermal
reservoir in the adiabatic regime.
