Realistically as long as the mortgage continues to be paid it is unlikely that the mortgage company would do anything. In the existing mortgage they likely have the right to call the mortgage if any of the original conditions change (primary residence, etc.) but as long as it is being paid it is unlikley they would do anything.
You could check with the mortgage holder about assuming the existing mortgage but there are very few truly easily assumable mortgages. Any assumption would require approval of the mortgage holder (likley much the same process as a mortgage application). My recommendation to a relative would be to simply continue to pay the mortgage – unless you can refinance into your name at a better rate/terms.