  We discuss and compare two recently proposed toy models for anomalous
transport and Griffiths effects in random systems near the Many-Body
Localization transitions: the random dephasing model, which adds thermal
inclusions in an Anderson Insulator as local Markovian dephasing channels that
heat up the system, and the random Gaussian Orthogonal Ensemble (GOE) approach
which models them in terms of ensembles of random regular graphs. For these two
settings we discuss and compare transport and dissipative properties and their
statistics. We show that both types of dissipation lead to similar
Griffiths-like phenomenology, with the GOE bath being less effective in
thermalising the system due to its finite bandwidth. We then extend these
models to the case of a quasi-periodic potential as described by the
Andr\'e-Aubry-Harper model coupled to random thermal inclusions, that we show
to display, for large strength of the quasiperiodic potential, a similar
phenomenology to the one of the purely random case. In particular, we show the
emergence of subdiffusive transport and broad statistics of the local density
of states, suggestive of Griffiths like effects arising from the interplay
between quasiperiodic localization and random coupling to the baths.
