Do you happen to know... can diplomatic immunity be shared with one’s children?
Yes, in some cases, it can be. "Official family members" of certain diplomats - but not all - share their diplomatic immunity.
(In simple terms, the family members of diplomats based at embassies share the diplomat's immunity, while the family members of those based at consulates or other centres don't (even though the diplomat themselves may still have immunity - but there are numerous exceptions and special cases to the rules).
Understand that if somebody with diplomatic immunity committed certain acts, it could constitute an act of war, even without being a crime. In such cases, you might expect most countries to waive somebody's immunity to try to avoid the consequences of committing an act of war.