The interdependent process of mind and matter is the stage of Dependent Origination where designation stops (DN 15). My personal understanding (that is not binding for anyone) is that the higher DO chain links found in so many other Suttas, namely sankhara (as khamma and volition) and ignorance are too weird to be adequately described with our language, because you can’t really wrap your head around how you should describe kammic formation and ignorance could be adequately described if we knew the Four Noble Truth; I would say we have no direct knowledge of them, so description of ignorance is always an approximation for us. Since among the khandhas at least sankharas are not included in vinnana, namarupa, and all lower DO chain links, we can conclude that it is at least the sankharas, i.e. kamma, that are shared between different lifetimes.
Now, if the overwhelming majority of people don’t have any memories of their previous lifetimes and most of the khandhas making up a human being get disrupted at the end of a lifetime (how many precisely remain untouched is a matter of debate, even though I bet on the kammic formations shaped by ignorance), it means the relationship with our future or past lives are not really close. One can always say one doesn’t associate themselves with those lives because they will have no or little memories of this current life. In other words, one can say there is so little conveyed from this lifetime into, say, the hell realm, that one may just as well conceive their ego as annihilated at their next death. For Buddhists, this view inevitably leads to more suffering, even if in this lifetime this annihilist is a very lovely and ethical person, for the simple reason that this atheist will be reborn, i.e. another being will be born due to their actions and cravings in this very life. Anyway, for a non-Buddhist my sankhara-oriented interpretation or rebirth and some interpretations of Parinibbana are pretty close to the atheist doctrine of annihilation, fair enough. In fact, in AN 10.29 the Buddha specifically praised annihilationism as the foremost wrong view.
For the record, this is all conjectural on my part, even though I presume everyone would agree at kamma being preserved between lifetimes, and I think other users can and most probably will present different explanations