In epiphenomenalist substance dualism, what can be a mechanism for how the physical affects the mental without invoking divine intervention?
Epiphenomenalist dualism is where the physical can affect the mental but not the other way around.
Epiphenomenalism, either substance or property epiphenomenalism, basically requires occasionalism -- that SOMETHING physical causes each and every mental event. Property epiphenomenalists have a candidate something -- that matter just has this property of metal events when it is in the appropriate configuration. Property epiphenomalists often try to come up with some general rule for this - David Chalmers postulates pan-psychism for all of matter -- and that inert things (rocks, thermostats) would therefore have a mental life. Frank Jackson just leaves it as an unknown TBD. http://homepages.wmich.edu/~baldner/jacksonf12.pdf
I have found no proposals for what that something is among substance epiphenomenalists. Basically, they take Jackson's position, and leave it up to physics and neuroscience to eventually sort out what causes occasionalism to work. Here are two recent relevant discussions by epiphenomenalists: https://philosophynow.org/issues/81/Epiphenomenalism_Explained https://philpapers.org/bbs/thread.pl?tId=666 The author of the first link asserts that consciousness is just an accidental byproduct -- and as most brain events don't produce consciousness, and many that do produce inappropriate consciousness -- he is committed to a thesis that the brain events that produce consciousness, and the consciousness events themselves being appropriate for the brian events -- is just a massive series of accidents that just happen to correspond.
William Robinson tries to address this point -- but all he comes up with is that he can assume occasionalism for both qualia AND choices, and that both dualism and physicalism have problems too -- http://www.public.iastate.edu/~wsrob/EvoEpi.pdf
Jaegwon Kim appears to be stuck with substance epiphenomenalism in Physicalism or Something Near Enough -- as he admits that some qualia are not physical. But it is only the "non-functional" qualia that he concedes are non-phyiscal -- and being non-functional -- they would not be investigatable or characterizable no matter how long neuroscientists studied the brain. Hence Kim is committed to mysterianism relative to qualia.
All of these are incoherent positions. The William James/Karl Popper objection that consciousness is FINELY TUNED by evolution is clearly true, and evolution cannot tune something that is not causal. https://plato.stanford.edu/entries/epiphenomenalism/ Robinson's occasionalism is not 100% successful (some events not conscious, some consciousnes is inappropriate), as one would expect for a biologically variable phenomena like consciousness, and his thesis requires that it IS 100% coupled to the physical.
Kim's position -- that SOME of consciousness is physical, and SOME isn't -- is bizarre. And his claim that qualia are non-functional is simply false -- our awareness of them shows "function", just not physical function.
The critiques of the epiphenomenalist position in the Philpapers dialog -- that epiphenomenalists must assert that their own knowledge of qualia is not caused by qualia -- and qualia are therefore totally causally decoupled from the physical -- applies decisively against Kim. The objections noted that our entire inference to a physical world is based on ASSUMING this causal correlation -- hence any acceptance of the physical as real (inference from phenomenology based on the principle of empirical indirect realism) requires rejection of epiphenomenalism from the outset.