All perfect praise be to Allah; and may His blessings and peace be upon Prophet Mohammad.
A person who broke the fast of one day of Ramadan for a lawful excuse, which continued after Ramadan until he died before making up for it, is free from liability, and his guardians are neither required to fast on his behalf nor to give an expiation (compensation for not fasting).
Al-Khateeb Ash-Shirbini (May Allah bless his soul) said: “There is no liability on the person who missed some days of Ramadan, but couldn`t make up for them because his sickness or lawful travel continued until his passing away. Therefore, neither fasting nor giving an expiation is due on him. Moreover, he isn`t considered sinful as he couldn`t perform an obligatory act of worship until he passed away, and this case is analogous to Hajj.“ {Al-Majmou`,vol.2/pp.172)}. And Allah Knows Best.