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.