So have you personally experienced this moment-to-moment rebirth thing? I'd be interested to hear what it's like, practically speaking.
Yeah, I not infrequently have the feeling of having just been "born" a second before. I obviously have memories and such, which is what allows me to make sense of my surroundings, but on occasion I do become aware that my sense of self is a continually arising phenomenon rather than a continuous state of being. It only seems like the latter most of the time because one of the functions of our ego-sense is to obscure the gaps by sort of filling them in, in much the same way that our brains fill in the gaps in our visual senses to give the impression that we have this continuous panoramic view of what's in front of us (we don't really).
Our teacher has called these "Zen moments" and encouraged us to cultivate them, provided we're not presently doing something that requires intense concentration, such as driving a car. He suggested that having several in a day would be a healthy thing to remind us how the mind actually works (and thus reduce self-attachment to mental functions, I suppose).
I think people do have this feeling sometimes but don't know what to make of it or don't pay attention and stop the mental chatter long enough to recognize it. Surely meditation practice helps. It's a feeling in which you could easily believe that you'd just been created a second ago, complete with memories. You're aware of what was going on just before that but don't feel a particular attachment to it. Everything seems new and kind of impressive, even though intellectually you know it's the same as it was before. It's actually a very subtle feeling, but I think it's interesting, so I've focused on it pretty intently when it has come up.