If you purchase a property and inherit the tenants, tenants who most likely paid security deposits at one point, how is the legal transfer of the security deposit money handled?

What you expected is irrelevant, the only thing that matters is what your purchase contract states.Normally you would demand a prorated rent, no deductions, as well as any and all deposits the tenants have paid.

They may be under the p false impression that their deposit can count towards the last month of rent.  

I'd send them the security deposit statement with move out damages and unpaid rent amounts and then at least you've done your part.

They want security that you have the down and enough to cover the holdings costs and what you'all need for deposits on materials like cabinets/counters/appliances until your bank draws kick in.

Personally, I set up a bank account for each of my tenants at a nearby bank and tell them to deposit the money into that account.  

The only way I see to cash flow is to put down a large deposit. 

Was there a security deposit?  

Some of the properties I'm looking at have sizable deposits and I'm trying to figure out if there is a chance I will need to sit on these for an extended length of time.  

After 30 weeks and all of those deposits I will have just under 10k.