What is passed from life to life in the process of rebirth? As a practising Buddhist, I am often asked this question: what is passed from life to life in the process of rebirth?

I have several answers to this depending on my audience - e.g. lessons to be learned, experiences to be resolved or, simply, one's own personal cockpit-style 'black box' of all that has been seen, heard, experienced etc in previous lives.

To what extent do colleagues believe that 'personality' or 'behavioural tendencies' should also be included?

I welcome your views. My question is directed at any practitioner of Buddhism, Theravadan or otherwise.
In the Pali suttas, I have not read anything about a "life to life" process of rebirth. In the Pali language, the word for "life" is "jiva". Where as the words often erroneously translated as "past lives" are "pubbe nivāsa", which literally mean "previous homes" &amp; is often more accurately translated as "past abodes", as follows: 


  Ye hi keci, bhikkhave, samaṇā vā brāhmaṇā vā anekavihitaṃ pubbenivāsaṃ anussaramānā anussaranti sabbete
  pañcu­pādā­nak­khan­dhe anussaranti etesaṃ vā aññataraṃ
  
  Bhikkhus, those ascetics and brahmins who recollect their manifold past abodes all recollect the five aggregates subject to clinging
  or a certain one among them.
  
  Any kind of aggregate, whether past, future or present, internal or external, gross or subtle, inferior or superior, far or near, all
  should be seen as it really is with correct wisdom thus: ‘This is not
  mine, this I am not, this is not my self.’
  
  SN 22.79 translation Bhikkhu Bodhi


Therefore, the many suttas that mention the Buddha recollecting his "past lives" are actually about the Buddha recollecting his "past homes/abodes", which, in respect to suttas such as the   Hāliddikāni Sutta &amp; SN 22.79, may have an entirely different meaning to "past life times". 

The above said, the general principles in the Pali suttas refer to the "death" ("marana") of one "body" ("kaya") and, if enlightenment does not occur, the "taking up" of (attaching/clinging to) another "body" ("kaya"). 

The words "marana" &amp; "kaya" do not necessarily mean "physical death" &amp; the "physical body". "Kaya", which also means "collection" or "group", can refer to a certain set of five aggregates (e.g. five aggregates affected by alcohol, which manifest as a happy person or happy being/satta). When those five aggregates or that state of 'being' ('satta'; refer to SN 23.2) ceases, one "kaya" &amp; "satta" dies ("marana) &amp; a new "kaya" &amp; "satta" arises (e.g. a new collection of five aggregates &amp; a new state of being that have a headache &amp; sickness or addiction due to drinking too much alcohol). 

In conclusion, in the same way an addiction to alcohol is developed, what passes from one "body" ("kaya"; "collection") to another "body" ("kaya"; "collection") is craving &amp; its associated underlying tendencies/habits ("anusaya") &amp; effects ("vipaka"). The short answer to the question is: "craving". 

Thus the Pali suttas state: 


  Vaccha, when a being sets this body aside and is not yet reborn in another body, I designate it as craving-sustained, for craving is its
  sustenance at that time.
  
  Kutuhalasala Sutta