Praise be to Allah.
If a Muslim dies owing expiation for breaking an oath, then his next of kin (the heirs) must offer expiation from his estate before it is divided. The expiation for breaking an oath is to free a slave, or feed ten poor persons, or clothe them. For a detailed explanation of that, please see question no. 45676.
They should choose the least expensive option (which nowadays is feeding the poor), because the rights of the heirs are connected to the estate, so spending too much on offering this expiation will harm the heirs, unless they agree to choosing a better option; the matter is up to them.
It says in Mughni al-Muhtaaj (6/192):
If someone dies owing expiation, then what must be done is to offer from his estate the option that is lowest in cost. End quote.
If the deceased was poor and did not leave wealth behind, then the expiation that must be offered is a fast of three days. So it is mustahabb (recommended) for his heir to fast on his behalf. He may also feed one poor person on each day instead of fasting.
The scholars of the Permanent Committee for Iftaa’ were asked:
A man died owing ten days of fasting to make up for missed Ramadan fasts. He recovered in Shawwaal (from a sickness he had in Ramadan), but he was careless about making up what he owed. Can his heir fast on his behalf, or is the fasting of the heir only applicable in the case of vows and offering expiation?
It is prescribed for his next of kin to fast on his behalf the number of days that he did not fast, because the Prophet (blessings and peace of Allah be upon him) said: “If someone dies owing fasts, his next of kin should fast on his behalf.” This hadith is general in meaning and applies to Ramadan fasts, fasts in fulfilment of vows, and fasts done as expiation, according to the correct view.
End quote from Fataawa al-Lajnah ad-Daa’imah (9/263).
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Expiation is required in the case of accidental killing.… – If the person who owes expiation dies without having offered that expiation, then his next of kin should feed sixty poor persons on his behalf, because that is instead of the fasting that the deceased was unable to do; as he would feed the poor on his behalf with regard to Ramadan fasts that the deceased owes, this is more appropriate.
End quote. Majmoo‘ al-Fataawa (34/170).
Shaykh ‘Abdullah at-Tayyaar (may Allah preserve him) said:
If a person dies without having offered expiation for his oath, is it prescribed for his heir to offer expiation for his oath?
Answer: The scholars differed concerning this ruling, but the correct view – and Allah knows best – is that his next of kin must offer expiation on his behalf from his wealth. If the deceased had wealth, then it is obligatory for his next of kin to offer expiation on his behalf, by feeding or clothing the poor, or by freeing a slave. If he did not have any wealth, then his next of kin or someone else should fast on his behalf, according to the soundest scholarly view. As to whether that is obligatory or recommended, there is a difference of opinion among the scholars. End quote.
And Allah knows best.