Yes, that's the essence. I would also state that there isn't a self that is reborn, but the unresolved actions and intentions that find the opportunity for resolution. Then a new self is born around those.
It's important to know that Buddhism doesn't believe in an eternal self.
Because the rich aren't the problem. Class is. Class consciousness is the solution.
It might not be your carrot. Total liberation is apparently a pleasurable state. It also has the additional effects of not harming another being through our unresolved kamma through a rebirth.
Another way to show compassion is to move closer to liberation so that the transmitted kamma is of a lighter quality and inclined towards liberation. We regularly wish strangers well and even do good works for them in this lifetime. Sometimes our effects can eminstr past this life at least according to Buddhism.
I hope that helped.