Standard quantum open system descriptions are being revisited with the advent of quantum reservoir engineering that aims to control quantum dynamics via strong dissipation. Despite its tremendous success, a full understanding of the interplay between intrinsic system and engineered dissipation channels is lacking. In this work, we present our theoretical framework for constructing quantum master equations that self-consistently incorporate all dissipation channels. Our method allows us to analytically obtain the induced rates on a system influenced by an engineered reservoir, all the way from the deep dispersive to the resonant regimes. Notably, this broad parameter regime is inaccessible within standard approaches based on Lindblad or Hamiltonian perturbation theory. Surprisingly, we find significant modifications to the rates at the leading order in system-reservoir coupling within the Born-Markov regime. We also discuss how our approach can be extended to structured reservoirs with strong quantum correlations and, as a direct experimental signature, quantify dispersive readout induced reduction or enhancement in qubit lifetimes.