You want to ensure you do not put them n the deed until your death. Reason being, if you do it before, it will be considered a gift and they will inherit your basis in the property. Where as if you wait, it will be considered inheritance and they will get a step up in basis to the fair market value at the date of death.
Assuming, it's a death, it will be a straight transition from you to them. A deceased person's mortgage becomes the responsibility of the person inheriting the home. The heir has several options such as moving into the home and assuming the mortgage, buying out other heirs if they also inherited a portion of the property, or selling the house and using the proceeds to pay off the mortgage.
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