The short answer is that we do not know what happens to the baby white walkers.
The longer answer is that we do not know what happens to the baby white walkers, but we can speculate. I'll start off by quoting what we know:
- The Night King was created thousands of years ago during the war between the First Men and the Children of the Forest.
- The Children of the Forest created the first White Walker - The Night King.
- The Night King can create White Walkers from alive baby boys.
- There are now old looking bearded White Walkers.
As such we can deduce that the ageing process of the white walkers is a lot different to men as the Night King is still alive and doesn't appear to have grown older.
If we take all of this into account it would appear that only the Night King has created White Walkers and only from baby boys. This would mean that White Walkers do age but differently to men and as such we don't know for sure.
Alternatively, there could be an army of white crawlers out there and the bearded ones are just older men that have been turned by the Night King.
Found on Google here, clicks through to Pinterest. According to @Chris the original is from here on deviant art.