Offspring don't just inherit DNA; DNA always comes in a package.

I think some people think of offspring as just being a passing on of DNA. But what you get in nature isn't just DNA, offspring start off as the combining of two packages, ie the gamete cells of a male and female. This combined package goes on to divide and grow into the new organism, that is the offspring.
The zygote is kind of a black box, and the DNA is only part of the package.
Personally I think this black box also has components other than simply materialistic based ones..ie non-atomic based, chemical ones.
Maybe the zygote contains memories of some or all of its ancestors, right back to the first cells on Earth,
This could be one explanation as to why some people feel they have lived past lives.

Maybe some of the ancient memories from the first cells were about how a cell learnt to function; how it could divide etc, Or maybe the first cells already came with those memories....these memoires would be handed down through the generations, and obviously still be relevant in today's cells and organisms.
Cells still have to know how to function, and divide etc.