Transference is generally minor, so any characteristics in the colony mites tend to dominate once established - especially if the bees are pushing them in that direction.
I don't know how much what I will say here will be supported - its one of those things that seems bleddy obvious to me once the first part (as explained by Steve Riley is in place).
As I understand it, if (VHS, or similar traits) bees detect mites in brood cells and uncapping occurs, they interfere with mite reproduction.
The bees seem to uncap more frequently in cases where there are many young mites inside the cell.
This behavior suppresses the strains of mites that tend to have large families, while leaving strains with smaller families relatively unaffected.
(Note: Family size is an inheritable trait.)
Over time, this pressure results in a mite population dominated by strains that produce only one or two offspring per cycle—levels the bees can manage.
The real problem arises when strains of mites that produce four or six offspring per cycle become prevalent. These strains multiply exponentially, leading to a so-called 'mite explosion' that overwhelms the bees.
I think you'll find that much in Steve Riley and the scientific work.
In effect (in my words now), the bees are selectively breeding low-fertility (or low-fecundity) mites, which they can keep under control.
