We study the dynamics of heat flux in the thermalization process of a pair of identical quantum systems that interact dissipatively with a reservoir in a cascaded fashion. Despite that 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 nonexponential time behavior 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 behaviors. 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.