Adiabatic elimination is a model reduction technique commonly used by physicists to eliminate quickly dissipating components from quantum physics equations. We revisit this technique when the target non-dissipating component is driven by Hamiltonian actuation at a fast timescale. Following center manifold theory, we can still write reduced dynamics for the target component, but there may be new conditions to ensure that it takes the standard structure of quantum dynamics, i.e. evolution equations of Lindblad type and coordinate changes in Kraus map form. We here propose various approaches to recover a Lindblad form up to third-order terms: without conditions for finite-dimensional systems, and under some finite set conditions for infinite-dimensional systems.