To attach this to stability is to implicitly assume that what's important is that
the wealth itself was passed down between generations. This would require stability, because if the wealth was all lost in one event, the chain would be irrevocably broken.
The alternative view is that the people in these families have the personal characteristics that allow one to build wealth, and that those characteristics are passed down (either by some kind of conditioning, or genetically, or some combination). This doesn't require much stability at all, since the wealth can all be lost but the capacity to build it is not as long as the person survives.
Examples of the second view can be found in diverse examples like: Wealthy Irans who lost everything in the '79 revolution, moved to America, and rebuilt from nothing to become wealthy again, often not even speaking English to start. Jews after the Holocaust, who lost everything and rebuilt from nothing to now become >17% of global billionaires on 0.1% of global population. And so on.
Though both views are obviously in effect, given the massive instability present throughout history, I'm inclined to say the second causality is dominant over the long term, while the first is only effective over the short term (e.g. trust-fund babies who lose everything in one generation, pro sports players or lottery winners who get millions and lose it all within a few years, etc).
I think it's important to keep both causalities in mind.