Yes, if you change the form of the title from the name of an individual or individuals, many banks will consider it to be a transfer and feel that their interest is no longer secured and call it "due on sale." So, whether you transfer it by quitclaim or warranty deed it makes no difference. What you need to do is get consent of whomever is financing the real estate agrees to honor the transfer or refinance the mortgage under the name of the LLC. Go through a real estate or business attorney.
The information provided in this answer does not create an attorney-client relationship and is not considered to be legal advice.
You cannot legally "sell" or "transfer" your property to the Family LLC without violating at least two provisions of the mortgage. There is no legal way around this. The lender will eventually learn you did this and will take legal action. Yes, you can refinance in the LLC name. But... most lenders consider lending money to an LLC for rental property extremely high risk. Lenders will lend money, but they usually want 65% Loan to Equity value and charge a much higher interest rate. What you want is the cheaper rate of a individual owner, while maintaining the protection of an LLC. No lender will do that. So go to your lender, tell them your plan, and get the property refinanced correctly in the LLC name or get ready to face the consequences when the lender learns you just transfer title on its collateral.
Answering this quetion does not establish an attorney client relation. The answer is for educational purposes only. You should consult an attorney for your circumstances.
Yes, you should refinance in the name of the LLC, although the lender will probably want personal guarantees on the loan. Since the purpose of the LLC is to create a legitimate business that is shielded as much as possible from creditors, the more steps you take to have theh LLC stand on its own, the better. As to the type of title to pass, the lender will probably have its own requirements.
This response is not intended to create an attorney/client relationship between any individuals, and is not intended as specific legal advice.