My PC has 8 kids, with the eldest triplets being 19 now. None of the other PCs have any children.
You could easily hand wave several years to a decade between each book if you want to, accompanied by whatever amount of Kingdom building you want or need. Those are the major built-in pauses in the AP.
There is no real time limit in the AP concerning how fast or slow the Kingdom should grow, but I would give the players whatever amount of time they need to accomplish any story goals they're working on.
I didn't do a multi-generational thing but it would be fairly easy, I would think.
Give a few years between books 2 & 3 (have the players meet the neighbors to the east and west so that the coming books mean something). You could give some time between books 3 & 4 (maybe have Fort Drelev start off really well at the beginning of the AP but it's dwindled down for the book they're in due to raids and what-not, pretty much as written). Between books 4 & 5 would allow you to pass as much time as you would like, same as between books 5 & 6.
I like Morrigan's idea of having a kingdom turn taking 6 months. (Because, yeah, the kingdom gets ridiculously huge and I think the players would get bored of expanding for all that time.) And you could use the reason that, sure, by the rules they could keep growing stupidly fast, but there's not enough people o fill the buildings they're making.
I had 5 years pass between books 2 & 3, and might add in a smaller gap between books 4 & 5. Or maybe 5 & 6 (we just finished book 4). The issue with the big gap that I found was the kingdom turns; it's just a chore to do so many at once, even with some handy software to speed the process. That's why I'm thinking of having shorter gaps in the future.
The party's ruler has 2 kids, and a retired PC (now NPC) has 1. They're still pretty young yet, though. Running the AP with generations of PCs would be kinda like making new, higher-level PCs for each adventure, or wherever you put the generational breaks. The increasing level structure of the AP works against the concept in that way. I do think the generations idea would be great for "the continuing adventures of <kingdom name>", though. Play through the AP with the original group of PCs and then come back X years later as their kids or grandkids take the stage with whole new adventures. That would be a neat way to go if the players became really attached to the kingdom.
I had quite a bit of fun with the rulers child.... I increased the amount of green haired items throughout the stolen land so when the baby had green eyes that sure got their attention for a little bit. The better payoff, however, was (out of game) months later and they realized that the child's first word, briar, actually had some meaning behind it.