"The new loan amount can be no more than the actual documented amount of the borrower's initial investment in purchasing the property plus the financing of closing costs, prepaid fees, and points on the new mortgage loan (subject to the maximum LTV, CLTV, and HCLTV ratios for the cash-out transaction based on the current appraised value).

To clarify, I'm asking how the prepaid rent would be transferred to the new landlord.

Yes prepaid taxes & insurance is the normal practice.

Disclaimer: I work for a rent payment serviceHi @Vitaliy Volpov , for the tenants that don't have a bank account we recommend they get a prepaid card that allows ACH debits, such as NetSpend, and they can use that like a bank account with an online payment service.  

I like the idea of a short term lease if it is entirely pre paid and with a firm date of departure.No more than 90 days,don't let them get lazy and comfortable.

If it were me, I'd buy the note with contract of sale for no out of pocket cash to noteholder and push all possible risks onto them, using my skills to foreclose, clear title, capitalize, repair and prepaid for retail resale.To sweeten pot, offer listing to him if he's halfway competent agent (have my doubts). 

A cash-out refinance within six (6) months of a purchase transaction when no financing was obtained for the purchase transaction are allowed under the following parameters: The new loan amount is not more than the actual documented amount of the borrower’s initial investment in purchasing the property, plus the financing of closing costs, prepaid fees, and points (subject to the maximum LTV).FOR AN INVESTMENT FIXED RATE CONVENTIONAL MORTGAGE; 1.

@Mike McGeeI know that they don't accept pre-paid or stolen card information.

Where I used to live the washers and dryers were operated using a laundry card, much like a prepaid credit card.  

That would allow you to look at duplex houses up to $77.2k (not including prepaids).