This question is best understood by contemplating a single lifetime to begin with, before extending that understanding to past and future lives.
This is how I do it.
The infant becomes the child who becomes the teenager who becomes a young adult and finally.. Here "I’’ am!
I look at the photograph of the teenager and say “Well, that was me.” 
Why?
The body, feeling tones, perceptions, volitional thoughts and consciousness I have today are not the same as those of that teenager. Let that sink in. The aggregates do not remain the same even from morning to night, let alone year to year (SN 12.62).
But I don’t feel or think that I change!
I think I am this present body and I think that I was that past body… I think “though the body may have changed, but its still mine”… I identify the changing bodily processual phenomena as a unitary Thing that has Form as an intrinsic quality. This is sakkayaditthi. And craving is the seamstress (AN 6.61).
Its like looking at a rainbow or a whirlpool or a flame and thinking that it has a form that is everlasting (nicca)… that is enjoyable (sukkha).. that can be possessed for all eternity or destroyed forever… (atta).
The basis for this is the craving and attachment that I have to be.. to exist. I find various grounds to base that existence on…one or the other of the 5 aggregates. And I delight in that experience. I never look at how I came to be.. instead I insist unwisely that I am this, I was that, I will be that (MN 2)… when pressed for proof I might point at my experience of the senses (vedana and sanna), or memory (sanna and sankhara) or thoughts and awareness (sankhara and vinnana)… etc.
This “I am” is processual too (AN 3.32).. its just a process of “I and mine making” occurring because of craving and ignorance.
I cannot conceive what I would be without these aggregates. So even if I do not think that I am these aggregates, I still hold to that whiff of something that I must be (SN 22.89)… this is the conceit of Self.
And so now it might be said that “I” am this process.. this mindstream.. this volition and intention.. kamma.
You see, I am still holding on. And that holding on is the very reason that these dependently originated processes continue. And these processes are the reason that the I and mine making keeps recurring… its an infinite loop!
There is an escape though. And the Buddha found it (SN 22.27).
One needs to see the Suffering in all this, to understand that this suffering has craving for existence at its heart, to let go of that craving and experience the cessation of Suffering by developing the 8 fold path… these are the 4 noble truths.
Then one can travel down the path of Disillusionment, Dispassion, Cessation, Peace, Insight, Awakening and Extinguishment (SN 47.32, SN 12.23).
A person is like a flame burning with the support of fuel (SN 44.9). The final extinguishment is likened to a flame going out (MN 72). And “I” could be likened to the keeper of the eternal flame (SN 35.28)!
What is it that would be extinguished? Its what is being brought into being and passed on from moment to moment, day to day, year to year, life to life.
Well what is that?
Simply, Dukkha samudaya (SN 12.51, SN 12.66).