The accurate description of non-adiabatic processes where the electronic and nuclear motion is highly correlated is fundamental to our understanding of a wide variety of phenomena ranging from photosynthesis , DNA photo-damage or charge transfer in photovoltaic cells . The full quantum-mechanical treatment of the dynamics beyond the Born-Oppenheimer approximation remains limited to a few degrees of freedom and approximate methods that treat the nuclei quantum-mechanically become unfeasible for large systems without a heavy investment in computational resources, or recent cutting-edge technologies . Justified by the large nuclear-electron mass ratio, trajectory-based schemes are the basis for many approximations for coupled electron-nuclear dynamics, where one propagates an ensemble of classical-like nuclear trajectories, each associated with a set of quantum electronic coefficients of, typically, a Born-Huang basis. The methods stem from a quantum evolution equation such as the time-dependent Schrödinger equation in the conventional Born-Huang picture , the exact factorization equations or the Liouville equation . In those frameworks, the classical limit for the nuclear motion is taken, while the electronic degrees of freedom are kept quantum-mechanical. Different trajectory-based schemes involve different approximations, and as a result of those approximations some exact physical constraints might be broken. One important example is the conservation of total energy which is a manifestation of the symmetry of the system under time-translation. Specifically, in the full quantum-mechanical picture the expectation value of the molecular Hamiltonian over the total molecular wavefunction is a constant of motion when no external fields are present. In the approximate trajectory-based view, energy can be exchanged between the ensemble of classical (nuclei) and quantum (electrons) degrees of freedom but should be such that the total energy of the system remains invariant. The most widely used trajectorybased methods, Ehrenfest dynamics and surface-hopping , impose this conservation law although in a somewhat restrictive way: each individual trajectory in the ensemble conserves its classical energy in these schemes but it is the energy of the trajectory ensemble as a whole that should be conserved, as it is the ensemble that represents the nuclear wavepacket. In principle, trajectories should be able to exchange energy, but this is not possible with independent trajectory schemes such as these. This tight constraint on the energy conservation of the system might prevent opening channels for the classical trajectories in the energy landscape that are accessible in the quantum dynamics, leading to inaccuracies. Further, the disconnect between coherent electronic evolution and incoherent nuclear trajectory propagation in surface-hopping means that energy is only conserved when the potential energy of the system is defined in terms of the instantaneous active state energy rather than from the coherent electronic evolution. This will be further discussed in the next section.