In my experience, this is a common story as I have known lots of parents who tell the same tale. I experienced this with my son.
1. Baby is born on the big side, often overdue. (My DS was 8 lbs, 11 oz and about 10 days late.)
2. Baby starts crossing percentile lines downwards on the charts (WHO included) in the first few weeks or months. The baby is showing completely normal development in all other respects. Some Dr's. become alarmed, though, which stresses the parents. This happened to me.
3. Baby often settles on to a new lower growth curve that is also more in line with the parents. i.e. if your partner and you are both of average height and weight, it is unlikely that your baby will stay at the 90th percentile. It is simply not in his or her genes. Dr. becomes less alarmed when baby stops moving downward on the charts.
Now, in my case, my ds (this is my 2.5 yo, not my newborn) has settled onto a curve that puts him at about the 20th percentile for height and weight. This is a bit concerning because my husband is 6 ft tall and slim and I am average height at 5,5 and used to be on the slim side (2 babies in two years has put me more into the average weight category) If my son doubles his height at two, he won't end up much taller than I am, which is at least fiver inches smaller than all the men in our family, save my mum's dad.
It could be that my son managed to get the "wee" genes from my mum's side of the family. All other grandparents are average height or tall. It is more likely that my son has constitutional growth delay (this isn't a disorder... it is just a slower pattern of growth than average.) Late bloomers abound on both sides of our family, particularly amongst the boys.
My DD is growing MUCH faster than DS did, but she was also more than a pound smaller at birth. She was born at 7,8. was 12 lbs at her two month appt and I think she is about 13 lbs at 12 weeks.
It is useful to bear in mind that the charts (WHO included) are just averages. Lots of children follow different growth patterns. We want to make sure that our children are realizing their genetic potential, and if they have lots of energy, are developing normally and have good output in the diaper department, there is no reason to believe that they are in any way undernourished or "failing" to thrive. The exception is when children are barely growing at all... in which case there may be a medical issue such as a lack of growth hormone. This can be tested for.