When the Pali suttas say that it is not the "same" thing that is born and dies what do they mean? When the Pali suttas say that it is not the "same" bundle of psycho-physical properties that is born and dies what do they mean: do they mean that conventional desginators like "I" only refer to different concrete things (a baby, an old man) but they also mean something different? 

Specifically: can we say, conventionally or otherwise, that the old man or the baby have aged, or that the baby grew up into an old man?


  “Exactly so, venerable sir. As I understand the Dhamma taught by the
  Blessed One, it is the same consciousness that runs and wanders
  through the round of rebirths, not another.”
  
  “What is that consciousness, Sāti?”
  
  “Venerable sir, it is that which speaks and feels and experiences here
  and there the result of good and bad actions.”
  
  “Misguided man, to whom have you ever known me to teach the Dhamma in
  that way?


Hope I haven't misread it! He then goes on to describe dependent origination.
When the Pali suttas say that it is not the "same" bundle of psycho-physical properties that is born and dies


The words "birth" ("jati") and "death" ("marana') are defined in SN 12.2 as producing or imputing the idea of "a being" ("satta") upon the manifestations or changing appearances of the aggregates. 

What is important to understand is not so much the aggregates but how ideas of 'self', 'persons' or 'beings' are imputed upon the aggregates. 

For example, when you see "my mother", "my father", "my wife", "my friend", "my enemy", "my favourite pop star", etc; and Arahant, in the same situation, sees only aggregates, elements and sense objects. 


  Specifically: can we say, conventionally or otherwise, that the old man or the baby have aged, or that the baby grew up into an old man?


Convention is different than ultimate truth. In ultimate truth, only aggregates change &amp; cease. This is why there are so many suttas that says an arahant is not born, does not age, does not die (example, MN 140; SN 22.85; etc). 

When there is no idea or view of a "self", "person" or "being", the ideas of birth &amp; death do not arise.