Is it known how far and how long a mite can travel on its own? Obligate  
parasites are often pretty vulnerable without their hosts.
 
Yes, varroa can only live for a few days at most, away from a bee colony. They are very fast crawlers, however, which has always made me deeply skeptical of screened bottom boards. I feel that mites that fall off the cluster are probably old and dying and that viable mites would be able to crawl back into the hive in any case. 
 
* * *

One  other way that you get transfer is the simplest- a dying hive, dead
bees,  lots of varroa now without a host and looking for somewhere to  go.

I would just point out that varroa mites are unlikely to be conscious of this, and do not make specific choices. They are simply programmed by evolution to perform certain behaviors which get them out of dying colonies and into live ones. The mites that don't do this, die. Life selects successful behaviors, and eliminates the rest.

Pete

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