  We study the dynamics of heat flux in the thermalization process of a pair of
identical quantum system that interact dissipatively with a reservoir in a {\it
cascaded} fashion. Despite the open dynamics of the bipartite system S is
globally Lindbladian, one of the subsystems "sees" the reservoir in a state
modified by the interaction with the other subsystem and hence it undergoes a
non-Markovian dynamics. As a consequence, the heat flow exhibits a
non-exponential time behaviour which can greatly deviate from the case where
each party is independently coupled to the reservoir. We investigate both
thermal and correlated initial states of $S$ and show that the presence of
correlations at the beginning can considerably affect the heat flux rate. We
carry out our study in two paradigmatic cases -- a pair of harmonic oscillators
with a reservoir of bosonic modes and two qubits with a reservoir of fermionic
modes -- and compare the corresponding behaviours. In the case of qubits and
for initial thermal states, we find that the trace distance discord is at any
time interpretable as the correlated contribution to the total heat flux.
