Follow up. I also heard that many armies built the catapults on site. Is that true?

I'm not a historian, I just study history a bit.

The introduction of these Asian engines in Europe an in England was a big deal, obviously. A large amount of materials, time, men transportation and shipping, assembly, etc, went in to these things.

I like this passage from 'The Medieval Way of War' that describes the building of some of the early engines in some detail

You can see there, it's a big deal, it becomes a a big project to deal with these things. Not all were as grand as those described in that book in actual battle, but engines were a big deal as to labor and resources. They were the singular focus of castle warfare. Also, yes they were often built on site, they had to be, but this could involve materials from a good distance away. Going to get them, shipping/transport, etc, became logistical concerns. So it could involve a 'region' of some size to build one.

Now, this may be an issue for sub rules, but I know about the stones, and cannot find a source. I just looked. So I'm gonna answer, and if you guys need to zap this on rules, no hard feelings.

Tradesmen would follow armies. It was like a whale in the deep ocean and they were kind of like remora. The army had big needs, of every type, and they usually - like today - had all the money to spend. So there were men who followed the engines and made ammo for them. Of course this was not always the case, but 'feeding the army's needs' was a preferred way of getting by. I've seen documentaries on this, where they visited impromptu 'quarries' at battle sites. While the engines were assembled, the stone men would fan out for stones, and use other ammo if no stones were there. They launched all kinds of things into cities just to make life unpleasant too, so it was not always stones. This link covers the various types of ammo used a bit. As they cite there, yes, Genghis Khan's army flinging plague-infected corpses over walls to infect cities played a considerable, if nor key role in 'The Black Death'. I say that to cover - it wasn't just stones they threw. If the horses die? Say as a hypothetical? Fling it in there - makes their lives so pleasant, might foul their water, cause disease, etc.

Hope that helps.