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We discuss and compare two recently proposed toy models for anomaloustransport and Griffiths effects in random systems near the Many-BodyLocalization transitions: the random dephasing model, which adds thermalinclusions in an Anderson Insulator as local Markovian dephasing channels thatheat up the system, and the random Gaussian Orthogonal Ensemble (GOE) approachwhich models them in terms of ensembles of random regular graphs. For these twosettings we discuss and compare transport and dissipative properties and theirstatistics. We show that both types of dissipation lead to similarGriffiths-like phenomenology, with the GOE bath being less effective inthermalising the system due to its finite bandwidth. We then extend thesemodels to the case of a quasi-periodic potential as described by theAndr\'e-Aubry-Harper model coupled to random thermal inclusions, that we showto display, for large strength of the quasiperiodic potential, a similarphenomenology to the one of the purely random case. In particular, we show theemergence of subdiffusive transport and broad statistics of the local densityof states, suggestive of Griffiths like effects arising from the interplaybetween quasiperiodic localization and random coupling to the baths.

Destruction of Localization by Thermal Inclusions: Anomalous Transport and Griffiths Effects in the Anderson and Andr\'e-Aubry-Harper Models