There are two ways that I see this working:

1. The retailers allow some sort of database access or provide with exported product lists from their stores.

2. There are crawlers built that crawl the other stores for products and then add them to the database on that site.

The above answer is along the right lines I would guess. One of those is probably the way that the site is populated with products.

The cart itself will then be populated with products from the farfetch site (that have been put there by the crawler). At this point I shouldnt think the actual site with the inventory will know anything about it. Once the transaction is put through on the farfetch site they will then have software that will automatically place the orders on all the respective sites using their own account details but the shoppers shipping details.

I've done a little research and Farfetch's has a kind of virtual inventory that is populated (either by partner shops, or their own staff I don't know), so it's not what I thought it was.
@ Andrianhenry yes that what Keep.com is doing now apparently, but it's unsustainable and they'll have to find a way to automate.
However I've found that other companies like Keep.com's OneCart, Cosmic Cart and Shoppable have come out with what they call "universal carts", which is what I was thinking.

While a truly universal cart seems really tough to implement, I'm wondering how hard it would be to have your cart sync with inventories of partner sites? I suppose you would have to create a plugin (for all major carts) that each partner site would have to install in order to sync with you?

The easiest way to do it if a company was in partnership with another site where the site would have a cart for the original company would be if the company built an API to interface with the cart. By doing this the site could easily just implement a cart with the API which would provide direct access to the original companies database and inventory.

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China Millionaire