There's no set rule on how the new Don might treat the deceased Don's family. But the outlook isn't good for the family:
Mob bosses are seldom the owners of record of businesses, private property or other tangible assets. If their names were on deeds and records, they'd provide a roadmap for law enforcement to prosecute them for racketeering, tax evasion, RICO, etc. They hide assets, and often, the family doesn't know where the assets are--or the guys fronting for those assets will try to cover up after the Don dies. Smart Dons try to look "poor" on official records. But when they die, the family often is stuck with the reality that they're far from rich.
Mafia finances resemble a "pyramid scheme": money is kicked upstairs from underlings to each level of boss. If there's one thing Dons have in common, it's greed. I wouldn't count on the new Don feeling charitable toward the former Don's family. That's probably why many Dons assure that at least one of their sons is made in the Mafia, or connected to a Mob-run business. Even then, the son probably will be resented or even considered a danger by the new Don and his people.
Great life, the Mafia.