My understanding from reading here and several other sources is that mites won't climb back up a distance after falling through the SBB. Doesn't make sense to me but that's what I've read.
SHB must spend a portion of its life cycle in the ground. There are chemical insecticide
treatments for the ground around your yard. I'm going to try parasitic nematodes around mine. SHB will fly in from a distance using smell of the hive but why not treat? I'm going to be applying it and so is the only other beek I know within miles of here. Considering the lack of feral hives these days, it could reduce the chances. Then again, my girls have been very good at keeping them in check so far.
Another interesting thing I found last year was a few wax moths on top of my screened inner covers. The smell drew them in up there but they couldn't get in to the combs.