If you haven't closed I would recommend you have them sign an estoppel agreement presented by the current owner that confirms rates and prepaids (last month rent , security deposit). 

Yes, I've never had this situation, and it seems as though the would-be renter somehow thinks I find the idea of prepaid rent appealing enough for me to drop the rent price.

Maybe the tenant is trying to prove that she can pay a years' rent (yes, I'm being optimistic here, but not ignoring the potential pitfalls...)Say that you don't accept pre-paid rent for accounting and tax reasons, but you would be happy to see a "proof of funds" in the way of 2-3 months bank statements.

Also consider that this tenant has a tad bit lower credit score than I'd prefer but pre paid a couple of months worth of rent already. 

In that case, I will use the prepaid rent to cover the monthly rent loss but once I have the property rented and say I have a balance of 3 months of rent left, does the remaining prepaid rent money go back to the tenant, if I find them?

I then let the surplus bank as extra pre-paid rent as they get back on their feet.The small raise also becomes an expected annual thing.  

This includes turning over pre-paid rent and security deposits to the new owner.

-prepayment charge on a prepaid amount over 20% of the original mortgage in any 12 month period, can only be a max of 6 months advance interest on that amount (over 20% of original mortgage).https://leginfo.legislature.ca.gov/faces/codes_displaySection.xhtml?

We have our own team of contractors we call when necessary.If a PMs sole job is to collect rent, then a pre-paid self addressed envelop is a whole lot cheaper than the almost $100 per house we currently pay.

Without that, the tenant can claim the old landlord was holding a deposit (and now you are responsible to return it) or that he prepaid rent for the next 5 years and it isn't his problem the old landlord didn't tell you.