The absolute first thing is to establish the jurisdiction of this scenario. The answer will be vastly different if the jurisdiction is California/USA than if the jurisdiction were South Susan. No shade against South Sudan, but we are talking about criminal and civil law, so the details might be very different.

But supposing this is a jurisdiction that follows in the Anglo-American common law (such as California, and I’ll proceed using California as the setting), then we can make some generally-true statement, some of which confirm what you’re already written:

1. Criminal law exists to punish bad acts committed against society at-large
2. Criminal law can only punish the persons or entities which have committed an act or omission that is proscribed in law, and only those persons or entities within the territory
3. Dead people or dissolved corporations are beyond the reach of criminal law
4. The notion that the next-of-kin will “inherit” the criminal liability was abolished long ago; see US Constitution “Bill of Attainder” prohibition, and equivalent in other jurisdiction like the UK or Australia
5. Anyone that is still alive and collaborated to aid or supply the dead assailant can be pursued using criminal law
6. In parallel to the criminal law system, civil lawsuits can be filed against the remaining property of the dead assailant. This is known as the “estate” of that person, and the lawsuit would be captioned as "XYZ v the Estate of [dead assailant]"
7. A civil lawsuit can only win as much property as the respondent (ie person being sued) has, or any insurance policy they had which might apply, or any debt which was owed to the respondent at the time of their death.
8. Mass murder commonly result in civil lawsuits that do not obtain anywhere near the full amount to compensate for the victims’ families’ loss.
9. As a result, the target of civil lawsuits can be expanded to include adjacent parties, such as the manufacturer of the weapon or materials used, under a claim of product liability or something similar. This is not a guaranteed result, but they often have deeper pockets and good insurance policies.
10. Civil lawsuits can only bring a monetary compensation. The law cannot revive the dead, cannot erase or amend history, and cannot salve the void left when victims are removed from this world unjustly.

With all that said, the entire line of inquiry into the dead assailant’s will, or to their parent’s will, or anything like that, is entirely inapplicable. Children or parents do not inherit the sins of others, at least where criminal liability and civil lawsuits are concerned. Unless, of course, the parents participated somehow or willfully neglected a duty to report (very few of these exist in California, unless the victims were undoubtedly known to be children; see mandatory reporting laws). Thus, these other people cannot be sued nor criminally punished.

The other commenter correctly said that what we call the “justice system” is more accurately called “harm reduction”. That’s not wrong, but I would post that the crimimal law system is about harm reduction (nb: I do not endorse the carcereal state of imprisoning huge segments of the population, disproportionately by race), whereas civil lawsuits are about equity and compensation.

Both systems exist in tandem to prevent people from achieving a bloodier form of justice in the streets, like in days of yore: pistols at dawn, dueling in general, “bigger army” diplomacy, midnight slaughters of whole families, and other such unpleasantries. It’s definitely not perfect, and it needs reforms in many parts, but the structure serves a purpose and so far, it’s what we have and the best that we have.